Geometry of Type C2

Geometry of Type $C_{2}$

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 16 March 2013

Geometry of Type $C_{2}$

Let

J = [\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ - 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \end{matrix}] .

Then we define the symplectic group

S P_{4} = {M \in G L_{n} (ℂ) ∣ M^{T} J M = J} .

If we write $M = [\begin{matrix} A & B \\ C & D \end{matrix}],$ then the condition $M^{T} J M = J$ yields $B^{T} D - D^{T} B = 0 = A^{T} C - C^{T} A,$ and $A^{T} D - C^{T} B = I = D^{T} A - B^{T} C .$ Although it is not immediately obvious, $S P_{4} \subseteq S L_{4} .$

Let $G = {SP}_{4} (ℂ)$ and $𝔤 = 𝔰 𝔭_{4} .$ Each root space $𝔤_{α}$ in $𝔤$ is 1-dimensional, and we choose distinguished vectors $e_{α}$ in each one.

\begin{matrix} e_{α_{1}} & = & [\begin{matrix} 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 1 \\ 0 & 0 & 0 & 0 \end{matrix}] \\ e_{α_{2}} & = & [\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \\ e_{α_{1} + α_{2}} & = & [\begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \\ e_{2 α_{1} + α_{2}} & = & [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \\ e_{- α_{1}} & = & [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & - 1 & 0 \end{matrix}] \\ e_{- α_{2}} & = & [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}] \\ e_{- α_{1} - α_{2}} & = & [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}] \\ e_{- 2 α_{1} - α_{2}} & = & [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}] \end{matrix}

Let $G_{s s}$ be the set of semisimple elements of $G$ and let $𝒩$ be the set of nilpotent elements of $𝔤 .$ Then $G$ acts on the set

Λ = {(s, n) ∣ s \in G_{s s}, n \in 𝒩_{q}^{s}}, where 𝒩_{q}^{s} = {n \in 𝒩 ∣ {Ad}_{s} (n) = q^{2} n},

g \cdot (s, n) = (g s g^{- 1}, {Ad}_{g} (n)) .

For $α \in R,$ let $e_{α}$ be a non-zero element in the $α$ root space of $𝔤 .$ Let $D \subseteq G$ be the subgroup of diagonal matrices in $G .$ Then

\begin{matrix} ϕ : T & ⟶ & D \\ t & ⟼ & s_{t} = [\begin{matrix} t (X^{ω_{2} - ω_{1}}) & 0 & 0 & 0 \\ 0 & t (X^{ω_{1}}) & 0 & 0 \\ 0 & 0 & t (X^{ω_{1} - ω_{2}}) & 0 \\ 0 & 0 & 0 & t (X^{- ω_{1}}) \end{matrix}] \\ \begin{matrix} t (X^{ω_{1}}) & = & d_{2} \\ t (X^{ω_{2}}) & = & d_{1} d_{2} \end{matrix} & ⟵ & [\begin{matrix} d_{1} & 0 & 0 & 0 \\ 0 & d_{2} & 0 & 0 \\ 0 & 0 & d_{1}^{- 1} & 0 \\ 0 & 0 & 0 & d_{2}^{- 1} \end{matrix}], \end{matrix}

is a bijection satisfying

{Ad}_{s_{t}} \cdot e_{α} = t (X^{α}) e_{α}

for $α \in R .$ Since $W_{0} = N (D) / D,$ $W_{0}$ acts on $D,$ and two diagonal elements $s$ and $s'$ are in the same $G -orbit$ if and only if they are in the same $W_{0} -orbit.$ The map $ϕ$ is $W_{0} -equivariant,$ giving a bijection

\begin{matrix} \begin{matrix} θ : & {W_{0} -orbits on T} & ⟷ & {G -orbits on G_{s s}} \\ t & ⟼ & s_{t} . \end{matrix} & (3.9) \end{matrix}

The weight $t_{z, w, i}$ satisfies $t (X^{α_{1}}) = z$ and $t (X^{α_{2}}) = w,$ for $i = 1$ or 2, and $z, w \in ℂ^{\times} .$ Then define

d_{α_{1}} (z) = s_{t_{z, 1, 1}} and d_{α_{2}} (z) = s_{t_{1, z, 1}},

for $z \in ℂ,$ so that

d_{α_{i}} (z) \cdot e_{α_{i}} = z e_{α_{i}} and d_{α_{i}} (z) \cdot e_{α_{j}} = e_{α_{j}},

for $i, j = 1, 2$ and $i \neq j .$ (We could have used $t_{z, 1, 2}$ and $t_{z, 1, 2}$ in our definitions or $d_{α_{i}} (z),$ but all that matters is to have some elements of $D$ that satisfy 3.4.)

We begin by choosing a set of representatives for the orbits in $Λ / G$ for all values of $q .$ After this, we give the bijection between $Λ / G$ and the irreducible $\tilde{H} -modules.$

Theorem 3.19. If $q^{2}$ is not a primitive $ℓ th$ root of unity with $ℓ \leq 4$ then the following is a set of representatives of the orbits in $Λ / G .$

\begin{matrix} (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], 0), \\ ([\begin{matrix} - 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], 0), \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1 / 2} \end{matrix}], 0), z \neq 1, q^{\pm 2}, \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], 0), \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}}), \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}], 0), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}], e_{α_{1}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], 0), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - q^{- 1} \end{matrix}], 0), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & z & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & z \end{matrix}], 0), z \neq \pm 1, \pm q^{\pm 1}, q^{\pm 2}, \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}], 0), \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}], e_{α_{1}}), \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}], e_{α_{2}}), \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}], e_{α_{1}} + e_{α_{2}}), \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & q^{2} z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & q^{- 2} z^{- 1 / 2} \end{matrix}], 0), z \neq 1, q^{\pm 2}, q^{- 4}, q^{- 6}, \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & q^{2} z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & q^{- 2} z^{- 1 / 2} \end{matrix}], e_{α_{1}}), z \neq 1, q^{\pm 2}, q^{- 4}, q^{- 6}, \\ ([\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], 0), \\ ([\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}}), \\ ([\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}), \\ ([\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{2} + e_{2 α_{1} + α_{2}}), \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], 0), z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2}, \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], e_{α_{2}}), z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2}, \\ (\pm [\begin{matrix} w^{1 / 2} & 0 & 0 & 0 \\ 0 & z w^{1 / 2} & 0 & 0 \\ 0 & 0 & w^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1} w^{- 1 / 2} \end{matrix}], 0), t_{z, w} generic, z \neq - 1, \end{matrix}

and

\begin{matrix} ([\begin{matrix} w^{1 / 2} & 0 & 0 & 0 \\ 0 & - w^{1 / 2} & 0 & 0 \\ 0 & 0 & w^{- 1 / 2} & 0 \\ 0 & 0 & 0 & - w^{- 1 / 2} \end{matrix}], 0), t_{- 1, w} generic. \end{matrix}

Proof.

Given an element $(s, n) \in Λ,$ by 3.9, there is an element $(s', n')$ in the $G -orbit$ of $(s, n)$ such that $s'$ is diagonal and that $s' = s_{t}$ for some $t_{z, w}$ such that $t_{z, w}$ is in the set of $W_{0} -orbit$ representatives listed in (2.6). Then by Lemma 3.9, it is sufficient to describe the $C_{G} (s_{t}) -orbits$ in $𝔤_{q}^{s}$ for a set of representatives of possible central characters $t .$ Also, Theorem 3.6 shows that $𝔤_{q}^{s}$ is spanned by ${e_{α} ∣ t (X^{α}) = q^{2}} .$

Case 1: $t_{1, 1}, t_{- 1, 1}, t_{1, z}, t_{z, 1}, t_{z, w}$

For the semisimples corresponding to $t_{1, 1}, t_{- 1, 1}, t_{1, z}, t_{z, 1}, t_{z, w}, 𝔤_{q}^{s} = 0,$ so that 0 is the only nilpotent that can be paired with $s .$ Note that if $t ∣_{Q}$ takes the form $t_{- 1, w},$ then $- s_{t} = s_{s_{1} t},$ so that these characters are in the same $W_{0} -orbit.$ Hence only one pair $(s_{t}, 0)$ is listed, and it listed separately from the other pairs involving $t_{z, 1}$ or $t_{z, w} .$

Case 2: $t_{\pm q, 1}, t_{q^{2}, z}, t_{z, q^{2}}$

For the weights $t_{q^{2}, 1}, t_{\pm q, 1}, t_{q^{2}, z}, t_{z, q^{2}}, ∣ P (t) ∣ = 1,$ so that $𝔤_{q}^{s}$ is 1-dimensional. Let $a e_{α} \in 𝔤_{q}^{s_{t}}$ with $a \in ℂ^{\times} .$ Then

d_{α} (a^{- 1}) \cdot (s_{t}, a e_{α}) = (s_{t}, e_{α}) .

Then all the elements $(s, n) \in Λ$ with $s = s_{t}$ are $G -conjugate$ to $(s_{t}, 0)$ or $(s_{t}, e_{α}) .$ Note that $\pm s_{t_{- 1, q^{2}}}$ are in the same $W_{0}$ orbit, so the $\pm$ is deleted from these semisimples in the listing above.

Case 3: $s_{t}$ for $t_{q^{2}, 1}$

If $s = s_{t_{q^{2}, 1}},$ then $𝔤_{q}^{s} = ℂ e_{α_{1}} + ℂ e_{α_{1} + α_{2}},$ and $C_{G} (s)$ is generated by $D$ and $x_{\pm α_{2}} (c)$ for $c \in ℂ^{\times} .$ Let $a e_{α_{1}} + b e_{α_{1} + α_{2}} \in 𝔤_{q}^{s} .$ If $a, b \in ℂ^{\times}$ then

x_{α_{2}} (- b / a) d_{α_{1}} (a) \cdot e_{α_{1}} = a e_{α_{1}} + b e_{α_{2}},

so that $a e_{α_{1}} + b e_{α_{2}}$ is $C_{G} (s) -conjugate$ to $e_{α_{1}} .$ If $a = 0$ then

x_{α_{1}} (b) x_{- α_{1}} (- 1 / b) \cdot b e_{α_{1} + α_{2}} = e_{α_{1}},

and if $b = 0$ then

d_{α_{1}} (a^{- 1}) \cdot a e_{α_{1}} = e_{α_{1}} .

Then every element $(s, n) \in Λ$ with $s = s_{t}$ is conjugate to $(s_{t}, 0)$ or $(s_{t}, e_{α_{1}}) .$

Case 4: $t_{q^{2}, q^{2}}$

If $t = t_{q^{2}, q^{2}}$ then $𝔤_{q}^{s_{t}} = ℂ e_{α_{1}} + ℂ e_{α_{2}} .$ If $a b \neq 0$ then

\begin{matrix} d_{α_{1}} (a^{- 1}) d_{α_{2}} (b^{- 1}) \cdot (s_{t}, a e_{α_{1}} + b e_{α_{2}}) = (s_{t}, e_{α_{1}} + e_{α_{2}}), \\ d_{α_{1}} (a^{- 1}) \cdot (s_{t}, a e_{α_{1}}) = (s_{t}, e_{α_{1}}), and \\ d_{α_{2}} (b^{- 1}) \cdot (s_{t}, b e_{α_{2}}) = (s_{t}, e_{α_{2}}) . \end{matrix}

Hence every pair $(s_{t}, n)$ with $n \neq 0$ is conjugate to one of these three elements of $Λ,$ but since

C_{G} (s) = {\pm d_{α_{1}} (c) d_{α_{2}} (d) ∣ c, d \in ℂ^{\times}},

none of these three elements are conjugate to each other. Thus $(s_{t}, 0), (s_{t}, e_{α_{1}}), (s_{t}, e_{α_{2}}),$ and $(s_{t}, e_{α_{1}} + e_{α_{2}})$ are the representatives of the orbits of elements of $Λ$ with $s = s_{t} .$

Case 5: $t_{- 1, q^{2}}$

If $t = t_{- 1, q^{2}}$ then $𝔤_{q}^{s_{t}} = ℂ e_{α_{2}} + ℂ e_{2 α_{1} + α_{2}} .$ If $a b \neq 0$ then

\begin{matrix} d_{α_{2}} (a^{- 1}) d_{α_{1}} (\sqrt{b^{- 1} a}) \cdot (s_{t}, a e_{α_{2}} + b e_{2 α_{1} + α_{2}}) = (s_{t}, e_{α_{2}} + e_{2 α_{1} + α_{2}}), \\ d_{α_{2}} (a^{- 1}) \cdot (s_{t}, a e_{α_{2}}) = (s_{t}, e_{α_{2}}), and \\ d_{α_{2}} (b^{- 1}) \cdot (s_{t}, b e_{2 α_{1} + α_{2}}) = (s_{t}, e_{2 α_{1} + α_{2}}) . \end{matrix}

Hence every pair $(s_{t}, n)$ with $n \neq 0$ is conjugate to one of these three elements of $Λ .$ Since $C_{G} (s_{t})$ consists of exactly the diagonal matrices, the $C_{G} (s_{t}) -orbit$ of $n \in 𝔤_{q}^{s}$ is $ℂ n .$ Thus $(s_{t}, 0), (s_{t}, e_{α_{1}}), (s_{t}, e_{α_{2}}),$ and $(s_{t}, e_{α_{1}} + e_{α_{2}})$ are representatives of the orbits of elements of $Λ$ with $s = s_{t} .$ Since $\pm s_{t_{- 1, q^{2}}}$ are in the same orbit, the $\pm$ is deleted from the pairs involving this semisimple above.

Case 6: $t_{1, q^{2}}$

If $t = t_{1, q^{2}},$ then

𝔤_{q}^{s_{t}} = ℂ e_{α_{2}} + ℂ e_{α_{1} + α_{2}} + ℂ e_{2 α_{1} + α_{2}} .

and

C_{G} (s_{t}) = ⟨ x_{α_{1}} (c), \pm d_{α_{1}} (a), \pm d_{α_{2}} (b) ∣ a, b, c \in ℂ^{\times} ⟩ .

Let $f e_{α_{2}} + g e_{α_{1} + α_{2}} + h e_{2 α_{1} + α_{2}} \in 𝒩_{q}^{s}$ with $f, g, h \in ℂ .$ If $g f = 0$ then

x_{- α_{1}} (d) \cdot h e_{2 α_{1} + α_{2}} = h d^{2} e_{α_{2}} + h d e_{α_{1} + α_{2}} + h e_{2 α_{1} + α_{2}},

so we assume $g f \neq 0 .$

If $f h = g^{2}$ then

x_{α_{1}} (\frac{- g}{g d + f}) x_{- α_{1}} (d) \cdot (f e_{α_{2}} + g e_{α_{1} + α_{2}} + h e_{2 α_{1} + α_{2}}) = (f + 2 g d + h d^{2}) e_{α_{2}} .

If $f h \neq g^{2}$ then

x_{α_{1}} (\frac{- h}{2 \sqrt{g^{2} - f h}}) x_{- α_{1}} (\frac{- g + \sqrt{g^{2} - h f}}{h}) \cdot (f e_{α_{2}} + g e_{α_{1} + α_{2}} + h e_{2 α_{1} + α_{2}}) = (\sqrt{g^{2} - h f}) e_{α_{1} + α_{2}} .

But

\begin{matrix} d_{α_{2}} ({(f + 2 g d + h d^{2})}^{- 1}) \cdot (f + 2 g d + h d^{2}) e_{α_{2}} = e_{α_{2}} and \\ d_{α_{2}} ({(\sqrt{g^{2} - h f})}^{- 1}) \cdot (\sqrt{g^{2} - h f}) e_{α_{1} + α_{2}} = e_{α_{1} + α_{2}} . \end{matrix}

Thus the elements $(s, n)$ in $Λ$ with $s = s_{t}$ are represented by $(s_{t}, 0), (s_{t}, e_{α_{2}}),$ and $(s_{t}, e_{α_{1} + α_{2}}) .$

$□$

Theorem 3.20. If $q^{2}$ is a primitive 4th root of unity then the following is a set of representatives of the orbits in $Λ / G .$

\begin{matrix} (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], 0), \\ ([\begin{matrix} - 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], 0), \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1 / 2} \end{matrix}], 0), z \neq 1, q^{\pm 2}, \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], 0), \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}}), \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}], 0), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}], e_{α_{1}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], 0), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - q^{- 1} \end{matrix}], 0), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & z & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & z \end{matrix}], 0), z \neq \pm 1, \pm q^{\pm 1}, q^{\pm 2}, \\ ([\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}], 0), \\ ([\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}], e_{α_{1}}), \\ ([\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}], e_{α_{2}}), \\ ([\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}], e_{α_{1}} + e_{α_{2}}), \\ ([\begin{matrix} q^{- 3} & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{3} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}} + e_{2 α_{1} + α_{2}}), \\ ([\begin{matrix} q^{- 3} & 0 & 0 & 0 \\ 0 & q^{- 1} & 0 & 0 \\ 0 & 0 & q^{3} & 0 \\ 0 & 0 & 0 & q \end{matrix}], e_{α_{1}} + e_{α_{2}}), \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & q^{2} z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & q^{- 2} z^{- 1 / 2} \end{matrix}], 0), z \neq \pm 1, q^{\pm 2}, q^{- 4}, q^{- 6}, \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & q^{2} z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & q^{- 2} z^{- 1 / 2} \end{matrix}], e_{α_{1}}), z \neq \pm 1, q^{\pm 2}, q^{- 4}, q^{- 6}, \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], 0), z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2}, \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], e_{α_{2}}), z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2}, \\ (\pm [\begin{matrix} w^{1 / 2} & 0 & 0 & 0 \\ 0 & z w^{1 / 2} & 0 & 0 \\ 0 & 0 & w^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1} w^{- 1 / 2} \end{matrix}], 0), t_{z, w} generic, z \neq - 1, \end{matrix}

and

\begin{matrix} (\pm [\begin{matrix} w^{1 / 2} & 0 & 0 & 0 \\ 0 & - w^{1 / 2} & 0 & 0 \\ 0 & 0 & w^{- 1 / 2} & 0 \\ 0 & 0 & 0 & - w^{- 1 / 2} \end{matrix}], 0), t_{z, w} generic. \end{matrix}

Proof.

The proof of Theorem 3.19 applies to this case as well, with changes only necessary in Case 4 and for the semisimple $s_{t_{- 1, q^{2}}} .$ Since $t_{- 1, q^{2}}$ is in the same $W_{0} -orbit$ as $t_{q^{2}, q^{2}}, s_{t_{- 1, q^{2}}}$ is in the same $G -orbit$ as $s_{t_{q^{2}, q^{2}}}$ and thus $s_{t_{- 1, q^{2}}}$ is omitted from the list. Also, the two weights $t$ with $t ∣_{Q} = t_{q^{2}, q^{2}}$ are in the same $W_{0} -orbit,$ and thus the $\pm$ is omitted in the list of representatives.

Case 4: $t_{q^{2}, q^{2}}$

If $t = t_{q^{2}, q^{2}}$ then $𝔤_{q}^{s_{t}} = ℂ e_{α_{1}} + ℂ e_{α_{2}} + ℂ e_{- 2 α_{1} - α_{2}} .$

If $a b c \neq 0$ then $a e_{α_{1}} + b e_{α_{2}} + c e_{- 2 α_{1} - α_{2}}$ is not nilpotent, since its minimal polynomial in the standard representation is $x^{4} + a^{2} b c .$

Then

\begin{matrix} d_{α_{1}} (a^{- 1}) d_{α_{2}} (a^{2} b^{- 1}) \cdot (s_{t}, a e_{α_{1}} + b e_{- 2 α_{1} - α_{2}}) = (s_{t}, e_{α_{1}} + e_{α_{2}}), \\ d_{α_{1}} ({(a b)}^{- 1 / 2}) d_{α_{2}} (b^{- 1}) \cdot (s_{t}, a e_{- 2 α_{1} - α_{2}} + b e_{α_{2}}) = (s_{t}, e_{- 2 α_{1} - α_{2}} + e_{α_{2}}), \\ d_{α_{1}} (a^{- 1}) d_{α_{2}} (b^{- 1}) \cdot (s_{t}, a e_{α_{1}} + b e_{α_{2}}) = (s_{t}, e_{α_{1}} + e_{α_{2}}), \\ d_{α_{2}} (a^{- 1}) \cdot (s_{t}, a e_{- 2 α_{1} - α_{2}}) = (s_{t}, e_{- 2 α_{1} - α_{2}}), and \\ d_{α_{1}} (a^{- 1}) \cdot (s_{t}, a e_{α_{1}}) = (s_{t}, e_{α_{1}}), and \\ d_{α_{2}} (b^{- 1}) \cdot (s_{t}, b e_{α_{2}}) = (s_{t}, e_{α_{2}}) . \end{matrix}

Hence every pair $(s_{t}, n)$ with $n \neq 0$ is conjugate to one of these six elements of $Λ,$ but since

C_{G} (s) = {\pm d_{α_{1}} (c) d_{α_{2}} (d) ∣ c, d \in ℂ^{\times}},

none of these elements are conjugate to each other. Thus the pairs $(s_{t}, 0), (s_{t}, e_{α_{1}}), (s_{t}, e_{α_{2}}), (s_{t}, e_{- 2 α_{1} - α_{2}}), (s_{t}, e_{- 2 α_{1} - α_{2}} + e_{α_{2}}), (s_{t}, e_{α_{1}} + e_{α_{2}})$ and $(s_{t}, e_{α_{1}} + e_{- 2 α_{1} - α_{2}})$ are representatives of the orbits of elements of $Λ$ with $s = s_{t} .$

However, later calculations are made easier by choosing nilpotent elements in $𝒩^{+}$ as representatives. Then we note:

\begin{matrix} w_{2} w_{1} \cdot (s_{t}, e_{- 2 α_{1} - α_{2}}) = (s_{w_{2} w_{1} t}, e_{α_{2}}), \\ w_{2} w_{1} \cdot (s_{t}, e_{- 2 α_{1} - α_{2}} + e_{α_{2}}) = (s_{w_{2} w_{1} t}, e_{α_{2}} + e_{2 α_{1} + α_{2}}), \end{matrix}

and

w_{2} w_{1} w_{2} \cdot (s_{t}, e_{α_{1}} + e_{- 2 α_{1} - α_{2}}) = (s_{w_{1} w_{2} w_{1} t}, e_{α_{1}} + e_{α_{2}}) .

These are the representatives listed above.

$□$

Theorem 3.21. If $q^{2}$ is a primitive third root of unity then the following is a set of representatives of the orbits in $Λ / G .$

\begin{matrix} (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], 0), \\ ([\begin{matrix} - 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], 0), \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1 / 2} \end{matrix}], 0), z \neq 1, q^{\pm 2}, \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], 0), \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}}), \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}], 0), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}], e_{α_{1}}), \\ (\pm [\begin{matrix} q^{2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & q^{- 2} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], e_{α_{2}}), \\ (\pm [\begin{matrix} q^{- 2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & q^{2} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], e_{α_{1}} + e_{α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], 0), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & z & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & z^{- 1} \end{matrix}], 0), z \neq \pm 1, q^{\pm 1}, q^{\pm 2}, \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & q^{2} z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & q^{- 2} z^{- 1 / 2} \end{matrix}], 0), z \neq 1, q^{\pm 2}, q^{- 4}, q^{- 6}, \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & q^{2} z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & q^{- 2} z^{- 1 / 2} \end{matrix}], e_{α_{1}}), z \neq 1, q^{\pm 2}, q^{- 4}, q^{- 6}, \\ ([\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], 0), \\ ([\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}}), \\ ([\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}), \\ ([\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}} + e_{2 α_{1} + α_{2}}), \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], 0), z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2}, \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], e_{α_{2}}), z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2}, \\ (\pm [\begin{matrix} w^{1 / 2} & 0 & 0 & 0 \\ 0 & z w^{1 / 2} & 0 & 0 \\ 0 & 0 & w^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1} w^{- 1 / 2} \end{matrix}], 0), t_{z, w} generic, z \neq - 1, \\ (\pm [\begin{matrix} w^{1 / 2} & 0 & 0 & 0 \\ 0 & - w^{1 / 2} & 0 & 0 \\ 0 & 0 & w^{- 1 / 2} & 0 \\ 0 & 0 & 0 & - w^{- 1 / 2} \end{matrix}], 0), t_{z, w} generic. \end{matrix}

Proof.

The proof of Theorem 3.19 applies, with changes only necessary in cases 3 and 4, and omitting the characters $t_{- q, 1}$ and $t_{q^{2}, q^{2}}$ (Case 4), since these characters are in the same orbits as $t_{q^{2}, 1}$ and $t_{1, q^{2}},$ respectively.

Case 3: $s_{t}$ for $t_{q^{2}, 1}$

If $s = s_{t_{q^{2}, 1}}$ then $𝔤_{q}^{s}$ is spanned by $e_{α_{1}}, e_{α_{1} + α_{2}},$ and $e_{- 2 α_{1} - α_{2}},$ while $C_{G} (s)$ is generated by $D$ and $x_{\pm α_{2}} (c)$ for $c \in ℂ^{\times} .$ In particular, $C_{G} (s)$ contains $w_{2} (t) = x_{α_{2}} (t) x_{- α_{2}} (- t^{- 1}) x_{α_{2}} (t),$ a representative of $s_{2} \in W_{0}$ in $N (T) / T .$

Let $n = w e_{α_{1}} + x e_{α_{1} + α_{2}} + y e_{- 2 α_{1} - α_{2}} \in 𝔤_{q}^{s},$ which is nilpotent. Since $d_{α_{2}} (y) \cdot (w e_{α_{1}} + x e_{α_{1} + α_{2}} + y e_{- 2 α_{1} - α_{2}}) = w e_{α_{1}} + x e_{α_{1} + α_{2}} + e_{- 2 α_{1} - α_{2}},$ we assume that $y = 0$ or 1.

If $y = 0,$ then case 3 in the proof of Theorem 3.19 shows that either $n = 0$ or $n$ is $C_{G} (s) -conjugate$ to $e_{α_{1}} .$ Next assume $y = 1 .$ If $w, x \in ℂ^{\times}$ then

d_{α_{2}} (w) d_{α_{1}} (w^{- 1}) x_{α_{2}} (x / w) \cdot w e_{α_{1}} + x e_{α_{1} + α_{2}} + e_{- 2 α_{2} - α_{1}} = e_{α_{1}} + e_{- 2 α_{1} - α_{2}} .

If $w = 0$ then

x_{α_{1}} (x) x_{- α_{1}} (- 1 / x) \cdot x e_{α_{1} + α_{2}} + e_{- 2 α_{1} - α_{2}} = e_{α_{1}} + e_{- 2 α_{1} - α_{2}},

and if $x = 0$ then

d_{α_{1}} (w^{- 1}) \cdot w e_{α_{1}} + e_{- 2 α_{1} - α_{2}} = e_{α_{1}} + e_{- 2 α_{1} - α_{2}} .

Then every element $(s, n) \in Λ$ with $s = s_{t}$ is conjugate to $(s_{t}, 0), (s_{t}, e_{α_{1}}), (s_{t}, e_{- 2 α_{1} - α_{2}}),$ or $(s_{t}, e_{α_{1}} + e_{- 2 α_{1} - α_{2}}) .$

Later calculations are made easier if we choose representatives of $n$ in $𝒩^{+} .$ So we note:

w_{2} w_{1} \cdot (s_{t}, e_{- 2 α_{1} - α_{2}}) = (s_{w_{2} w_{1} t}, e_{α_{2}}),

and

w_{2} w_{1} w_{2} \cdot (s_{t}, e_{α_{1}} + e_{- 2 α_{1} - α_{2}}) = (s_{w_{2} w_{1} w_{2} t}, e_{α_{1}} + e_{α_{2}}) .

These are the representatives listed above.

$□$

Theorem 3.22. If $q^{2} = - 1$ then the following is a set of representatives of the orbits in $Λ / G .$

\begin{matrix} (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], 0), \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1 / 2} \end{matrix}], 0), z \neq 1, q^{\pm 2} \\ ([\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], 0), \\ ([\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}}), \\ ([\begin{matrix} q^{- 1} & 0 & 0 & 0 \\ 0 & q^{- 1} & 0 & 0 \\ 0 & 0 & q & 0 \\ 0 & 0 & 0 & q \end{matrix}], e_{α_{2}}), \\ ([\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{1} + α_{2}}), \\ ([\begin{matrix} q^{- 1} & 0 & 0 & 0 \\ 0 & q^{- 1} & 0 & 0 \\ 0 & 0 & q & 0 \\ 0 & 0 & 0 & q \end{matrix}], e_{α_{1} + α_{2}}), \\ ([\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{- 1} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q \end{matrix}], e_{α_{2}} + e_{α_{1}}), \\ ([\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{- 1} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q \end{matrix}], e_{2 α_{1} + α_{2}} + e_{α_{2}}), \\ ([\begin{matrix} q^{- 1} & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{1}} + e_{α_{2}}), \\ ([\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}], 0), \\ ([\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}], e_{α_{1}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], 0), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{- 1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q \end{matrix}], e_{2 α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & z & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & z \end{matrix}], 0), z \neq 1, q^{\pm 1}, q^{\pm 2}, \\ ([\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & q^{2} z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & q^{- 2} z^{- 1 / 2} \end{matrix}], 0), z \neq 1, q^{\pm 2}, q^{- 4}, q^{- 6}, \\ ([\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & q^{2} z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & q^{- 2} z^{- 1 / 2} \end{matrix}], e_{α_{1}}), z \neq 1, q^{\pm 2}, q^{- 4}, q^{- 6}, \\ ([\begin{matrix} q^{2} z^{1 / 2} & 0 & 0 & 0 \\ 0 & z^{1 / 2} & 0 & 0 \\ 0 & 0 & q^{- 2} z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1 / 2} \end{matrix}], e_{α_{1}}), z \neq 1, q^{\pm 2}, q^{- 4}, q^{- 6}, \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], 0), z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2}, \\ (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], e_{α_{2}}), z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2}, \\ (\pm [\begin{matrix} q^{- 1} & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], e_{α_{2}}), z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2}, \\ (\pm [\begin{matrix} w^{1 / 2} & 0 & 0 & 0 \\ 0 & z w^{1 / 2} & 0 & 0 \\ 0 & 0 & w^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1} w^{- 1 / 2} \end{matrix}], 0), t_{z, w} generic, z \neq - 1, \end{matrix}

and

\begin{matrix} (\pm [\begin{matrix} w^{1 / 2} & 0 & 0 & 0 \\ 0 & - w^{1 / 2} & 0 & 0 \\ 0 & 0 & w^{- 1 / 2} & 0 \\ 0 & 0 & 0 & - w^{- 1 / 2} \end{matrix}], 0), t_{- 1, w} generic. \end{matrix}

Proof.

The proof of Theorem 3.19 applies to this case as well, with essential changes only necessary in Cases 2, 3 and 4 and for the semisimple $s_{t_{q^{2}, q^{2}}} .$ Case 1 now covers the central characters $t_{1, 1}, t_{1, z}, t_{z, 1},$ and $t_{z, w} .$ Since $t_{q^{2}, q^{2}}$ is in the same $W_{0} -orbit$ as $t_{q^{2}, 1},$ $s_{t_{q^{2}, q^{2}}}$ is in the same $G -orbit$ as $s_{t_{q^{2}, 1}}$ and thus $s_{t_{q^{2}, q^{2}}}$ (Case 4) is omitted from the list.

Case 2: $s_{t}$ for $t_{q^{2}, z}, t_{z, q^{2}},$ or $t_{\pm q, 1}$

Since $P (t) = {\pm β},$ where $β$ is $α_{1}, α_{2},$ or $2 α_{1} + α_{2},$ $𝔤_{q}^{s}$ is spanned by $e_{\pm β} .$ However, $a e_{β} + b e_{- β}$ is nilpotent exactly when $a b = 0 .$ Since $C_{G} (s_{t}) = D,$ and

d_{α_{i}} (c) \cdot e_{\pm β} = \pm c ⟨ β, ω_{i} ⟩ e_{\pm β},

every element of $𝒩_{s}^{q}$ is $C_{G} (s_{t}) -conjugate$ to either $e_{β}$ or $e_{- β} .$

Case 3: $s_{t}$ for $t_{q^{2}, 1}$

If $s = s_{t_{q^{2}, 1}}$ then $𝔤_{q}^{s}$ is spanned by $e_{\pm α_{1}}$ and $e_{\pm (α_{1} + α_{2}) .}$ And $C_{G} (s)$ is generated by $D,$ $x_{\pm α_{2}} (c)$ and $x_{\pm (2 α_{1} + α_{2})}$ for $c \in ℂ^{\times} .$ In particular, $C_{G} (s)$ contains representatives of $s_{2}$ and $s_{1} s_{2} s_{1},$ and thus of $s_{2} s_{1} s_{2} s_{1}$ as well.

Let $n = f e_{α_{1}} + g e_{α_{1} + α_{2}} + h e_{- α_{1}} + j e_{- α_{1} - α_{2}} \in 𝔤_{q}^{s} .$ Then $n$ is nilpotent exactly if $h f + g j = 0,$ since its minimal polynomial in the standard representation of $𝔤$ is $x^{2} - (h f + g j) .$

By applying the argument in case 3 of Theorem 3.19, $n$ is $C_{G} (s)$ conjugate to an element of $𝔤_{q}^{s}$ with its positive part equal to either $e_{α_{1}}$ or 0. Thus we may assume $g = 0 .$ Then if $n$ is nilpotent, $h f = 0$ as well.

If $g = h = 0$ but $f j \neq 0,$ then

x_{- (2 α_{1} + α_{2})} (1) x_{2 α_{1} + α_{2}} (- \frac{f}{j} - 1) \cdot f e_{α_{1}} + j e_{- α_{1} - α_{2}} = - j e_{α_{1}} .

If $f = g = 0$ but $h j \neq 0,$ then

s_{2} s_{1} s_{2} s_{1} \cdot h e_{- α_{1}} + j e_{- α_{1} - α_{2}} = h e_{α_{1}} + j e_{α_{1} + α_{2}},

which is in the same $C_{G} (s) -orbit$ as $h e_{α_{1}} .$

If exactly one of $f, h, j$ is non-zero, then $n = a e_{β}$ for some $β \in {\pm α_{1}, \pm (α_{1} + α_{2})} .$ But the roots $\pm α_{1}$ and $\pm (α_{1} + α_{2})$ are in the same orbit under the action of $s_{2}$ and $s_{1} s_{2} s_{1} .$ Hence, $n$ is conjugate to $a e_{α_{1}},$ where $a$ is equal to the non-zero coefficient of $n .$

Finally,

d_{α_{1}} (f^{- 1}) \cdot f e_{α_{1}} = e_{α_{1}} .

Thus every element of $𝒩_{s}^{q}$ is in the same $C_{G} (s) -orbit$ as either $e_{α_{1}}$ or 0.

Case 4: $s_{t}$ for $t_{1, q^{2}}$

If $s = s_{t_{1, q^{2}}}$ then $𝔤_{q}^{s}$ is spanned by $e_{\pm α_{2}}, e_{\pm (α_{1} + α_{2})},$ and $e_{\pm (2 α_{1} + α_{2})},$ while $C_{G} (s)$ is generated by $D$ and $x_{\pm α_{1}} (c) .$

By the argument in case 4 of theorem 3.19, if $n \neq 0$ is in the span of $e_{α_{2}}, e_{α_{1} + α_{2}},$ and $e_{2 α_{1} + α_{2}},$ then $n$ is $C_{G} (s) -conjugate$ to either $e_{α_{2}}$ or $e_{α_{1} + α_{2}} .$ We also note that

C_{G} (s) \cdot ⟨ e_{α_{2}}, e_{α_{1} + α_{2}}, e_{2 α_{1} + α_{2}} ⟩ \subseteq ⟨ e_{α_{2}}, e_{α_{1} + α_{2}}, e_{2 α_{1} + α_{2}} ⟩,

since $C_{G} (s)$ is the same as in the generic case.

A similar argument using $x_{- α_{1}} (c)$ in place of $x_{α_{1}} (c)$ shows that if $n \neq 0$ is in the span of $e_{- α_{2}}, e_{- (α_{1} + α_{2})},$ and $e_{- (2 α_{1} + α_{2})},$ then $n$ is $C_{G} (s) -conjugate$ to either $e_{- α_{2}}$ or $e_{- (α_{1} + α_{2})} .$

We now examine the general case. Let $n = f e_{α_{2}} + g e_{α_{1} + α_{2}} + h e_{2 α_{1} + α_{2}} + j e_{- α_{2}} + k e_{- α_{1} - α_{2}} + l e_{- 2 α_{1} - α_{2}} .$ Assume that not all of $f, g, h$ are zero and not all of $j, k, l$ are zero, or else we are in one of the cases above. As noted above, $f e_{α_{2}} + g e_{α_{1} + α_{2}} + h e_{2 α_{1} + α_{2}}$ is $C_{G} (s) -conjugate$ to either $e_{α_{2}}$ or $e_{α_{1} + α_{2}} .$ Let $x \cdot (f e_{α_{2}} + g e_{α_{1} + α_{2}} + h e_{2 α_{1} + α_{2}}) = e_{α_{2}}$ or $e_{α_{1} + α_{2}},$ for some $x \in C_{G} (s) .$

Then because the positive and negative parts of $𝔤_{q}^{s}$ are fixed by the action of $C_{G} (s),$

x \cdot n = e_{α_{2}} + j' e_{- α_{2}} + k' e_{- α_{1} - α_{2}} + l' e_{- 2 α_{1} - α_{2}}

x \cdot n = e_{α_{1} + α_{2}} j' e_{- α_{2}} + k' e_{- α_{1} - α_{2}} + l' e_{- 2 α_{1} - α_{2}} .

Thus we can assume $f = 1$ and $g = h = 0,$ or $g = 1$ and $f = h = 0 .$

Case 1: $f = 1, g = h = 0$

If $n = e_{α_{2}} + j e_{- α_{2}} + k e_{- α_{1} - α_{2}} + l e_{- 2 α_{1} - α_{2}},$ then the minimal polynomial of $n$ in the defining representation of ${𝔰 𝔭}_{4}$ is $x^{4} - j x^{2} .$ Hence if $n$ is nilpotent, $j = 0 .$ Then if $k \neq 0,$

d_{α_{1}} (k) x_{- α_{l / (2 k)}} \cdot (e_{α_{2}} + k e_{- α_{1} - α_{2}} + l e_{- 2 α_{1} - α_{2}}) = e_{α_{2}} + e_{- α_{1} - α_{2}} .

If $k = 0,$ then

d_{α_{1}} (\sqrt{l}) \cdot e_{α_{2}} + l e_{- 2 α_{1} - α_{2}} = e_{α_{2}} + e_{- 2 α_{1} - α_{2}} .

Case 2: $g = 1, f = h = 0$

If $n = e_{α_{1} + α_{2}} + j e_{- α_{2}} + k e_{- α_{1} - α_{2}} + l e_{- 2 α_{1} - α_{2}},$ then the minimal polynomial of $n$ in the defining representation of ${𝔰 𝔭}_{4}$ is $x^{4} - 2 k + k^{2} - j l .$ Hence if $n$ is nilpotent, $k = 0$ and either $j$ or $l$ is zero.

If $j = 0,$ then

w_{1} \cdot (e_{α_{1} + α_{2}} + l e_{- 2 α_{1} - α_{2}}) = e_{α_{1} + α_{2}} + l e_{- α_{2}} .

Then

d_{α_{2}} (j) d_{α_{1}} (j^{- 1}) \cdot (e_{α_{1} + α_{2}} + j e_{- α_{2}}) = e_{α_{1} + α_{2}} + e_{- α_{2}} .

Thus, $n$ is in the $C_{G} (s)$ orbit of either $0,$ $e_{α_{2}},$ $e_{α_{1} + α_{2}},$ $e_{- α_{2}},$ $e_{- α_{1} - α_{2}},$ $e_{α_{2}} + e_{- α_{1} - α_{2}},$ $e_{α_{2}} + e_{- 2 α_{1} - α_{2}},$ or $e_{α_{1} + α_{2}} + e_{- α_{2}} .$

Future computations will be easier if we choose orbit representatives $(s, n)$ with $n \in 𝒩^{+} .$ Then we note:

\begin{matrix} w_{0} \cdot (s_{t}, e_{- α_{2}}) & = & (s_{w_{0} t}, e_{α_{2}}), \\ w_{0} \cdot (s_{t}, e_{- α_{1} - α_{2}}) & = & (s_{w_{0} t}, e_{α_{1} + α_{2}}), \\ w_{1} w_{2} w_{1} \cdot (s_{t}, e_{α_{2}} + e_{- α_{1} - α_{2}}) & = & (s_{w_{1} w_{2} w_{1} t}, e_{α_{2}} + e_{α_{1}}), \\ w_{1} w_{2} w_{1} \cdot (s_{t}, e_{α_{2}} + e_{- 2 α_{1} - α_{2}}) & = & (s_{w_{1} w_{2} W_{1} t}, e_{α_{2}} + e_{2 α_{1} + α_{2}}), \end{matrix}

and

\begin{matrix} w_{2} \cdot (s_{t}, e_{α_{1} + α_{2}} + e_{- α_{2}}) & = & (s_{w_{2} t}, e_{α_{1}} + e_{α_{2}}) . \end{matrix}

These are the orbit representatives listed above.

$□$

Theorem 3.23. If $q = - 1$ then the following is a set of representatives of the orbits in $Λ / G .$

\begin{matrix} (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], 0), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], e_{α_{1}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], e_{α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], e_{α_{1}} + e_{α_{2}}), \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1 / 2} \end{matrix}], 0), z \neq \pm 1, q^{\pm 2} \\ (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1 / 2} \end{matrix}], e_{α_{1}}), z \neq \pm 1, q^{\pm 2} \\ ([\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}], 0), \\ ([\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}], e_{2 α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}], e_{α_{2}} + e_{2 α_{1} + α_{2}}), \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & z & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & z \end{matrix}], 0), z \neq \pm 1, \\ (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & z & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & z \end{matrix}], e_{α_{2}}), z \neq \pm 1, \end{matrix}

and

\begin{matrix} (\pm [\begin{matrix} w^{1 / 2} & 0 & 0 & 0 \\ 0 & z w^{1 / 2} & 0 & 0 \\ 0 & 0 & w^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1} w^{- 1 / 2} \end{matrix}], 0), t_{z, w} generic. \end{matrix}

Proof.

By (3.9) and the classification of central characters given in 2.4.1, the semisimple elements listed above are a set of representatives of the semisimple orbits in $G .$ Then it suffices to show that the nilpotent elements paired with each $s_{t}$ are a set of representatives of the $C_{G} (s) -orbits$ in $𝒩_{q}^{s_{t}}$ For $t \in T,$

s_{t} = \pm d_{α_{1}} (t (X^{α_{1}})) d_{α_{2}} (t (X^{α_{2}})) .

The centralizer of $s_{t}$ in $G$ is generated by ${x_{\pm α} (c) ∣ c \in ℂ^{\times}, t (X^{α}) = 1}$ and $D .$

Case 1: $t_{1, 1}$

Since $s_{t_{1, 1}}$ is central, $C_{G} (s_{t}) = G$ and $𝒩_{s}^{q} = 𝒩 .$ Hence the nilpotent $C_{G} (s) -orbits$ in $𝒩_{s}^{q}$ are the nilpotent orbits in $𝔤 .$ These are represented by 0, $e_{α_{1}},$ $e_{α_{2}},$ and $e_{α_{1}} + e_{α_{2}} .$

Case 2: $t_{1, z}$ and $t_{z, 1}$

If $s = s_{t_{1, z}}$ or $s_{t_{z, 1}},$ then $P (t) = {e_{α_{i}}},$ where $i = 1$ or 2, respectively. In these cases, $C_{G} (s)$ is generated by $D$ and $x_{\pm α_{i}} (c),$ for $c \in ℂ^{\times} .$ In particular, $C_{G} (s)$ contains a representative of $w_{i} \in W_{0} .$

However, $a e_{α_{i}} + b e_{- α_{i}}$ is nilpotent exactly if $a b = 0 .$ If $a = 0,$ then

w_{i} \cdot b e_{- α_{i}} = b e_{α_{i}},

so we assume that $n = a e_{α_{i}} .$

Then

d_{α_{i}} (a^{- 1}) \cdot a e_{α_{i}} = e_{α_{i}},

so that every non-zero element of $𝒩_{q}^{s}$ is $C_{G} (s) -conjugate$ to $e_{α_{i}} .$

Case 3: $t_{q, 1}$

If $s = s_{t_{q, 1}},$ then $P (t) = {α_{1}, 2 α_{1} + α_{2}}$ and $𝔤_{q}^{s}$ is spanned by $e_{\pm (2 α_{1} + α_{2})}$ and $e_{\pm α_{2}} .$ Also, $C_{G} (s)$ is generated by $D$ and $x_{\pm α_{2}} (c)$ and $x_{\pm (2 α_{1} + α_{2})} (c)$ for $c \in ℂ^{\times} .$ In particular, $C_{G} (s)$ contains a representative of $w_{2}$ and $w_{1} w_{2} w_{1} .$

However, $a e_{α_{2}} + b e_{- α_{2}} + c e_{2 α_{1} + α_{2}} + d e_{- 2 α_{1} - α_{2}}$ is nilpotent exactly if $a b = 0$ and $c d = 0 .$ However,

w_{2} \cdot (b e_{- α_{2}} + c e_{2 α_{1} + α_{2}} + d e_{- 2 α_{1} - α_{2}}) = b e_{α_{2}} + c e_{2 α_{1} + α_{2}} + d e_{- 2 α_{1} - α_{2}},

and

w_{1} w_{2} w_{1} \cdot (a e_{α_{2}} + b e_{- α_{2}} + d e_{- 2 α_{1} - α_{2}}) = a e_{α_{2}} + b e_{- α_{2}} + d e_{2 α_{1} + α_{2}},

so we can assume $n = a e_{α_{2}} + c e_{2 α_{1} + α_{2}} .$ Then if $a c \neq 0,$

d_{α_{1}} (\sqrt{a / c}) d_{α_{2}} (a^{- 1}) \cdot (a e_{α_{2}} + c e_{2 α_{1} + α_{2}}) = e_{α_{2}} + e_{2 α_{1} + α_{2}} .

If $a = 0$ but $c \neq 0,$ then

d_{α_{2}} (c^{- 1}) \cdot c e_{2 α_{1} + α_{2}} = e_{2 α_{1} + α_{2}} .

if $c = 0$ but $a \neq 0$ then

d_{α_{2}} (a^{- 1}) \cdot a e_{α_{2}} = e_{α_{2}} .

Hence $n$ is in the $C_{G} (s) -orbit$ of either 0, $e_{α_{2}},$ or $e_{α_{2}} + e_{2 α_{1} + α_{2}} .$

Case 4: $t_{z, w}$

If $t = t_{z, w},$ then $P (t) = \emptyset,$ and $𝒩_{q}^{s} = 0 .$

$□$

Cosets

The irreducible $\tilde{H} -modules$ can also be constructed as the Borel-Moore homology of generalized Springer fibers. If $s$ is a semisimple element of $G$ and $n \in 𝔤_{q}^{s},$ then $ℬ_{(s, n)}$ is the set of Borel subalgebras fixed by $s$ and containing $n .$ Identifying $G / B$ with $ℬ$ shows that $ℬ_{(s, n)}$ is the set of $B -cosets$ in $G$ fixed by the action of $s$ and $exp (n) .$ Thus, it is necessary to compute this action of $G$ on cosets in $G / B .$

The group $G$ is generated by:

H_{α} (c) = w_{α} (c) w_{α} (- 1) and x_{α} (c) = exp (c e_{α}), for α \in R,

with some added relations (see [Ste1967]).

The first relations are commutator relations between the $x_{α} (c) .$

\begin{matrix} x_{α_{1}} (c) x_{α_{2}} (d) = x_{α_{2}} (d) x_{α_{1}} (c) x_{α_{1} + α_{2}} (c d) x_{2 α_{1} + α_{2}} (- c^{2} d) \\ x_{α_{1}} (c) x_{α_{1} + α_{2}} (d) = x_{α_{1} + α_{2}} (d) x_{α_{1}} (c) x_{2 α_{1} + α_{2}} (2 c d) \\ x_{α_{1}} (c) x_{2 α_{1} + α_{2}} (d) = x_{2 α_{1} + α_{2}} (d) x_{α_{1}} (c) \\ x_{α_{1}} (c) x_{- α_{2}} (d) = x_{- α_{2}} (d) x_{α_{1}} (c) \\ x_{α_{1}} (c) x_{- α_{1} - α_{2}} (d) = x_{- α_{1} - α_{2}} (d) x_{α_{1}} (c) x_{- α_{2}} (- 2 c d) \\ x_{α_{1}} (c) x_{- 2 α_{1} - α_{2}} (d) = x_{- 2 α_{1} - α_{2}} (d) x_{α_{1}} (c) x_{- α_{1} - α_{2}} (- c d) x_{- α_{2}} (- c^{2} d) \\ x_{- α_{1}} (c) x_{α_{1}} (d) = x_{- α_{1}} (c) x_{α_{1}} (- c^{- 1}) x_{α_{1}} (d + c^{- 1}) = x_{α_{1}} (c^{- 1}) w_{1} x_{α_{1}} (d + c^{- 1}) \\ x_{α_{2}} (c) x_{α_{1} + α_{2}} (d) = x_{α_{1} + α_{2}} (d) x_{α_{2}} (c) \\ x_{α_{2}} (c) x_{2 α_{1} + α_{2}} (d) = x_{2 α_{1} + α_{2}} (d) x_{α_{2}} (c) \\ x_{α_{2}} (c) x_{- α_{1} - α_{2}} (d) = x_{- α_{1} - α_{2}} (d) x_{α_{2}} (c) x_{- α_{1}} (c d) x_{- 2 α_{1} - α_{2}} (- d^{2} c) \\ x_{α_{2}} (c) x_{- 2 α_{1} - α_{2}} (d) = x_{- 2 α_{1} - α_{2}} (d) x_{α_{2}} (c) \\ x_{- α_{2}} (c) x_{α_{2}} (d) = x_{- α_{2}} (c) x_{α_{2}} (- c^{- 1}) x_{α_{2}} (d + c^{- 1}) = x_{α_{2}} (c^{- 1}) w_{2} x_{α_{2}} (d + c^{- 1}) \end{matrix}

The next relations describe how the Weyl group interacts with the $x_{α} (c) .$

\begin{matrix} w_{1} x_{α} (c) w_{1}^{- 1} = x_{s_{1} (α)} (- c) if α = \pm (α_{1} + α_{2}) or \pm α_{1} . \\ w_{1} x_{α} (c) w_{1}^{- 1} = x_{s_{1} (α)} (c) if α = \pm α_{2} or \pm (2 α_{1} + α_{2}) . \\ w_{1} x_{\pm (α_{1} + α_{2})} (c) w_{1}^{- 1} = x_{\pm (α_{1} + α_{2})} (c) . \\ w_{2} x_{α} (c) w_{2}^{- 1} = x_{s_{2} (α)} (- c) if α = \pm α_{1} or \pm α_{2} . \\ w_{2} x_{α} (c) w_{2}^{- 1} = x_{s_{2} (α)} (c) if α = \pm (α_{1} + α_{2}) or \pm (2 α_{1} + α_{2}) . \end{matrix}

Explicitly, $G$ is generated by

\begin{matrix} x_{α_{1}} (c) = [\begin{matrix} 1 & 0 & 0 & 0 \\ c & 1 & 0 & 0 \\ 0 & 0 & 1 & - c \\ 0 & 0 & 0 & 1 \end{matrix}] x_{α_{2}} (c) = [\begin{matrix} 1 & 0 & c & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], \\ x_{α_{1} + α_{2}} (c) = [\begin{matrix} 1 & 0 & 0 & c \\ 0 & 1 & c & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] x_{2 α_{1} + α_{2}} (c) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & c \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], \\ x_{- α_{1}} (c) = [\begin{matrix} 1 & c & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & - c & 1 \end{matrix}] x_{- α_{2}} (c) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ c & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], and \\ x_{α_{1} + α_{2}} (c) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & c & 1 & 0 \\ c & 0 & 0 & 1 \end{matrix}] x_{2 α_{1} + α_{2}} (c) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & c & 0 & 1 \end{matrix}] . \end{matrix}

The Varieties $ℬ_{s}$

We use Theorem 3.8 to determine the varieties $ℬ_{s}$ for semisimple elements $s \in D .$ The following semisimple elements are defined only up to an element of $Z (G),$ but this does not affect $ℬ_{s} .$

First,

s_{t_{1, 1}} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] = H_{α_{1}} (1) H_{α_{2}} (1) \in ℤ (G),

so that $ℬ_{s_{t_{1, 1}}} = ℬ .$

Next,

s_{t_{- 1, 1}} = [\begin{matrix} - 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] = s_{t_{- 1, 1}} = H_{α_{2}} (- 1) .

Then $ℬ_{s_{t_{- 1, 1}}}$ consists of $B,$ $w_{1} B,$ $x_{α_{2}} (c) w_{2} B,$ $w_{1} x_{α_{2}} (c) w_{2} B,$ $x_{α_{2}} (c) w_{2} w_{1} B,$ $w_{1} x_{α_{2}} (c) w_{2} w_{1} B,$ $x_{α_{2}} (c) w_{2} w_{1} x_{α_{2}} (d) w_{2} B,$ and $x_{α_{2}} (c) w_{2} w_{1} x_{α_{2}} (d) w_{2} w_{1} B,$ for $c, d \in ℂ .$

If $z \neq 1, q^{\pm 2} z \neq 1, q^{\pm 2},$ then

s_{t_{1, z}} = [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1 / 2} \end{matrix}] = H_{α_{1}} (z^{1 / 2}) H_{α_{2}} (z) .

Then $ℬ_{s_{t_{1, z}}}$ consists of $B,$ $w_{2} B,$ $x_{α_{1}} (c) w_{1} B,$ $w_{2} w_{1} B,$ $x_{α_{1}} (c) w_{1} w_{2} B,$ $w_{2} w_{1} w_{2} B,$ $x_{α_{1}} (c) w_{2} w_{1} w_{2} B,$ and $x_{α_{1}} (c) w_{1} w_{2} w_{1} w_{2} B,$ for $c \in ℂ .$ Geometrically, this consists of four disjoint copies of $ℙ^{1} .$

Next,

s_{t_{1, q^{2}}} = [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}] = H_{α_{1}} (q) H_{α_{2}} (q^{2}) .

Then $ℬ_{s_{t_{1, q^{2}}}}$ contains $B,$ $w_{2} B$ $x_{α_{1}} (c) w_{1} B,$ $w_{2} w_{1} B,$ $x_{α_{1}} (c) w_{1} w_{2} B,$ $w_{2} w_{1} w_{2} B,$ $x_{α_{1}} (c) w_{1} w_{2} w_{1} B,$ and $x_{α_{1}} (c) w_{1} w_{2} w_{1} w_{2} B,$ for $c \in ℂ .$ (This changes for specific values of $q$ only if $q^{2} = 1,$ in which case $s_{t_{1, q^{2}}}$ is central.) Geometrically, this consists of four disjoint copies of $ℙ^{1}$

Next,

s_{t_{q^{2}, 1}} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}] = H_{α_{1}} (q^{2}) H_{α_{2}} (q^{2}) .

Then $ℬ_{s_{t_{q^{2}, 1}}}$ contains $B,$ $w_{1} B,$ $x_{α_{2}} (c) w_{2} B,$ $w_{1} w_{2} B,$ $x_{α_{2}} (c) w_{2} w_{1} B,$ $w_{1} w_{2} w_{1} B,$ $x_{α_{2}} (c) w_{2} w_{1} w_{2} B,$ and $w_{1} w_{2} w_{1} x_{α_{2}} (c) w_{2} B,$ for $c \in ℂ .$ (This changes only if $q^{2} = \pm 1,$ in which case $s_{t}$ was computed above.) Geometrically, this consists of four disjoint copies of $ℙ^{1} .$

Next,

s_{t_{q, 1}} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}] = H_{α_{1}} (q) H_{α_{2}} (q) .

Then $ℬ_{s_{t_{q, 1}}}$ consists of $B,$ $w_{1} B,$ $x_{α_{2}} (c) w_{2} B,$ $w_{1} w_{2} B,$ $x_{α_{2}} (c) w_{2} w_{1} B,$ $w_{1} w_{2} w_{1} B,$ $x_{α_{2}} (c) w_{2} w_{1} w_{2} B,$ and $w_{1} w_{2} w_{1} x_{α_{2}} (c) w_{2} B,$ for $c \in ℂ .$ (This changes only if $q = \pm 1,$ in which case $s_{t}$ was computed above.) Geometrically, this consists of four disjoint copies of $ℙ^{1} .$

Next,

s_{t_{- q, 1}} = \pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - q^{- 1} \end{matrix}] = H_{α_{1}} (- q) H_{α_{2}} (- q) .

Then $ℬ_{s_{t_{- q, 1}}}$ contains $B,$ $w_{1} B,$ $x_{α_{2}} (c) w_{2} B,$ $w_{1} w_{2} B,$ $x_{α_{2}} (c) w_{2} w_{1} B,$ $w_{1} w_{2} w_{1} B,$ $x_{α_{2}} (c) w_{2} w_{1} w_{2} B,$ and $w_{1} w_{2} w_{1} x_{α_{2}} (c) w_{2} B,$ for $c \in ℂ .$ (This changes only if $q = \pm 1,$ in which case $s_{t}$ was computed above.) Geometrically, this is four disjoint copies of $ℙ^{1} .$

If $z \neq 1, q^{\pm 1}, q^{\pm 2},$ then

s_{t_{z, 1}} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & z & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & z \end{matrix}] = s_{t_{z, 1}} = H_{α_{1}} (z) H_{α_{2}} (z) .

Then $ℬ_{s_{t_{1, q^{2}}}}$ contains $B,$ $w_{1} B,$ $x_{α_{2}} (c) w_{2} B,$ $w_{1} w_{2} B,$ $x_{α_{2}} (c) w_{2} w_{1} B,$ $w_{1} w_{2} w_{1} B,$ $x_{α_{2}} (c) w_{2} w_{1} w_{2} B,$ and $w_{1} w_{2} w_{1} x_{α_{2}} (c) w_{2} B,$ for $c \in ℂ .$ Geometrically, this is four disjoint copies of $ℙ^{1} .$

Next,

s_{t_{q^{2}, q^{2}}} = [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}] = H_{α_{1}} (q^{4}) H_{α_{2}} (q^{3}) .

Then $ℬ_{s_{t_{q^{2}, q^{2}}}}$ consists of $w B$ for $w \in W_{0},$ eight disjoint points.

This changes if $q^{4} = 1,$ $q^{6} = 1,$ or $q^{2} = 1 .$ If $q^{2} = 1,$ then $s_{t}$ is central. If $q^{4} = 1,$ then $s_{t}$ is in the same orbit as $s_{t_{1, q^{2}}}$ above. If $q^{6} = 1,$ then $ℬ_{s_{t_{q^{2}, q^{2}}}}$ consists of $w B$ for $w \in W_{0},$ $w_{1} x_{α_{2}} (c) w_{2} B,$ $w_{1} x_{α_{2}} (c) w_{2} w_{1} B,$ $w_{1} x_{α_{2}} (c) w_{2} w_{1} w_{2} B,$ and $w_{2} w_{1} x_{α_{2}} (c) w_{2} B .$ Geometrically, this is four disjoint copies of $ℙ^{1} .$

If $z \neq 1, q^{\pm 2}, q^{- 4}, q^{- 6},$ then

s_{t_{q^{2}, z}} = [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & q^{2} z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & q^{- 2} z^{- 1 / 2} \end{matrix}] .

Then $ℬ_{s_{t_{q^{2}, z}}}$ consists of $w B$ for $w \in W_{0},$ which is eight disjoint points.

Next,

s_{t_{- 1, q^{2}}} = [\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}] = H_{α_{1}} (q) H_{α_{2}} (- q^{2}) .

Then $ℬ_{s_{t_{- 1, q^{2}}}}$ consists of $w B$ for $w \in W_{0},$ which is eight disjoint points.

If $z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2},$ then

s_{t_{z, q^{2}}} = [\begin{matrix} q & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}] = H_{α_{1}} (z q) H_{α_{2}} (z q^{2}) .

Then $ℬ_{s_{t_{z, q^{2}}}}$ consists of $w B$ for $w \in W_{0},$ which is eight disjoint points.

Nilpotent Elements

The next goal is to understand the action of $exp (c e_{α})$ on the elements of $G / B .$

Case: $exp (c e_{α_{1}}) = x_{α_{1}} (c) :$

Cosets starting with $x_{α_{1}} (d) :$

\begin{matrix} x_{α_{1}} (c) \cdot x_{α_{1}} (d) w_{1} B & = & x_{α_{1}} (c + d) w_{1} B, \\ x_{α_{1}} (c) \cdot x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} B & = & x_{α_{1}} (c + d) w_{1} x_{α_{2}} (e) w_{2} B, \\ x_{α_{1}} (c) \cdot x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} B & = & x_{α_{1}} (c + d) w_{1} x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} B, \\ x_{α_{1}} (c) \cdot x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} x_{α_{2}} (g) w_{2} B & = & x_{α_{1}} (c + d) w_{1} x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} x_{α_{2}} (g) w_{2} B, \end{matrix}

Cosets starting with $x_{α_{2}} (d) :$

\begin{matrix} x_{α_{1}} (c) \cdot x_{α_{2}} (d) w_{2} B & = & x_{α_{2}} (d) x_{α_{1}} (c) x_{α_{1} + α_{2}} (c d) x_{2 α_{1} + α_{2}} (- c^{2} d) w_{2} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1} + α_{2}} (- c) x_{α_{1}} (c d) x_{2 α_{1} + α_{2}} (- c^{2} d) B \\ = & x_{α_{2}} (d) w_{2} B \\ x_{α_{1}} (c) \cdot x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} B & = & x_{α_{2}} (d) x_{α_{1}} (c) x_{α_{1} + α_{2}} (c d) x_{2 α_{1} + α_{2}} (- c^{2} d) w_{2} x_{α_{1}} (e) w_{1} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1} + α_{2}} (- c) x_{α_{1}} (c d) x_{2 α_{1} + α_{2}} (- c^{2} d) x_{α_{1}} (e) w_{1} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1} + α_{2}} (- c) x_{α_{1}} (c d + e) x_{2 α_{1} + α_{2}} (- c^{2} d) w_{1} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c d + e) x_{α_{1} + α_{2}} (- c) x_{2 α_{1} + α_{2}} (c^{2} d + 2 c e) w_{1} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c d + e) w_{1} x_{α_{1} + α_{2}} (c) x_{α_{1}} (c^{2} d + 2 c e) B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c d + e) w_{1} B \\ x_{α_{1}} (c) \cdot x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) B & = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c d + e) w_{1} x_{α_{1} + α_{2}} (c) x_{α_{1}} (c^{2} d + 2 c e) x_{α_{2}} (f) w_{2} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c d + e) w_{1} x_{α_{2}} (f) x_{α_{1} + α_{2}} (c) x_{α_{1}} (c^{2} d + 2 c e) \\ \cdot x_{α_{1} + α_{2}} (c^{2} d f + 2 c e f) x_{2 α_{1} + α_{2}} (- f {(c^{2} d + 2 c e)}^{2}) w_{2} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c d + e) w_{1} x_{α_{2}} (f) w_{2} x_{α_{1}} (c) x_{α_{1} + α_{2}} (- c^{2} d - 2 c e) \\ \cdot x_{α_{1}} (+ c^{2} d f + 2 c e f) x_{2 α_{1} + α_{2}} (- f {(+ c^{2} d + 2 c e)}^{2}) B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c d + e) w_{1} x_{α_{2}} (f) w_{2} B \end{matrix}

Case: $exp (c e_{α_{2}}) = x_{α_{2}} (c) :$

Cosets starting with $x_{α_{2}} (d) :$

\begin{matrix} x_{α_{2}} (c) \cdot x_{α_{2}} (d) w_{2} B & = & x_{α_{2}} (c + d) w_{2} B \\ x_{α_{2}} (c) \cdot x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} B & = & x_{α_{2}} (c + d) w_{2} x_{α_{1}} (e) w_{1} B \\ x_{α_{2}} (c) \cdot x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B & = & x_{α_{2}} (c + d) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B \\ x_{α_{2}} (c) \cdot x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} x_{α_{1}} (g) w_{1} B & = & x_{α_{2}} (c + d) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} x_{α_{1}} (g) w_{1} B \end{matrix}

Cosets starting with $x_{α_{1}} (d) :$

\begin{matrix} x_{α_{2}} (c) \cdot x_{α_{1}} (d) w_{1} B & = & x_{α_{1}} (d) x_{α_{2}} (c) x_{α_{1} + α_{2}} (- c d) x_{2 α_{1} + α_{2}} (c^{2} d) w_{1} B \\ = & x_{α_{1}} (d) w_{1} x_{α_{1} + α_{2}} (c) x_{2 α_{1} + α_{2}} (c d) x_{α_{2}} (c^{2} d) B \\ = & x_{α_{1}} (d) w_{1} B \\ x_{α_{2}} (c) \cdot x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} B & = & x_{α_{1}} (d) w_{1} x_{2 α_{1} + α_{2}} (c) x_{α_{1} + α_{2}} (c d) x_{α_{2}} (c^{2} d) x_{α_{2}} (e) w_{2} B \\ = & x_{α_{1}} (d) w_{1} x_{α_{2}} (c^{2} d + e) w_{2} x_{2 α_{1} + α_{2}} (c) x_{α_{1}} (c d) B \\ = & x_{α_{1}} (d) w_{1} x_{α_{2}} (c^{2} d + e) w_{2} B \\ x_{α_{2}} (c) \cdot x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} B & = & x_{α_{1}} (d) w_{1} x_{α_{2}} (c^{2} d + e) w_{2} x_{2 α_{1} + α_{2}} (c) x_{α_{1}} (c d) x_{α_{1}} (d) w_{1} B \\ = & x_{α_{1}} (d) w_{1} x_{α_{2}} (c^{2} d + e) w_{2} x_{α_{1}} (f + c d) w_{1} x_{α_{2}} (c) B \\ = & x_{α_{1}} (d) w_{1} x_{α_{2}} (c^{2} d + e) w_{2} x_{α_{1}} (f + c d) w_{1} B \end{matrix}

Case: $exp (c e_{α_{1} + α_{2}}) = x_{α_{1} + α_{2}} (c) :$

Cosets starting with $x_{α_{1}} (d) :$

\begin{matrix} x_{α_{1} + α_{2}} (c) \cdot x_{α_{1}} (d) w_{1} B & = & x_{α_{1}} (d) x_{α_{1} + α_{2}} (c) x_{2 α_{1} + α_{2}} (- 2 c d) w_{1} B \\ = & x_{α_{1}} (d) w_{1} x_{α_{1} + α_{2}} (- c) x_{α_{2}} (- 2 c d) B \\ = & x_{α_{1}} (d) w_{1} B \\ x_{α_{1} + α_{2}} (c) \cdot x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} B & = & x_{α_{1}} (d) w_{1} x_{α_{1} + α_{2}} (c) x_{α_{2}} (- 2 c d) x_{α_{2}} (e) w_{2} B \\ = & x_{α_{1}} (d) w_{1} x_{α_{1} + α_{2}} (c) x_{α_{2}} (e - 2 c d) w_{2} B \\ = & x_{α_{1}} (d) w_{1} x_{α_{2}} (e - 2 c d) w_{2} x_{α_{1}} (c) B \\ = & x_{α_{1}} (d) w_{1} x_{α_{2}} (e - 2 c d) w_{2} B \\ x_{α_{1} + α_{2}} (c) \cdot x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} B & = & x_{α_{1}} (d) w_{1} x_{α_{2}} (e - 2 c d) w_{2} x_{α_{1}} (c) x_{α_{1}} (f) w_{1} B \\ = & x_{α_{1}} (d) w_{1} x_{α_{2}} (e - 2 c d) w_{2} x_{α_{1}} (c + f) w_{1} B \end{matrix}

Cosets starting with $x_{α_{2}} (d) :$

\begin{matrix} x_{α_{1} + α_{2}} (c) \cdot x_{α_{2}} (d) w_{2} B & = & x_{α_{2}} (d) x_{α_{1} + α_{2}} (c) w_{2} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c) B \\ = & x_{α_{2}} (d) w_{2} B \\ x_{α_{1} + α_{2}} (c) \cdot x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} B & = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c) x_{α_{1}} (e) w_{1} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c + e) w_{1} B \\ x_{α_{1} + α_{2}} (c) \cdot x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B & = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c) x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1}} (c + e) w_{1} x_{α_{2}} (f) w_{2} B \\ x_{α_{1} + α_{2}} (c) \cdot x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} x_{α_{1}} (g) w_{1} B & = & x_{α_{2}} (d) w_{2} α_{1} (c) x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} x_{α_{1}} (g) w_{1} B \\ = & x_{α_{2}} (d) w_{2} α_{1} (c + e) w_{1} x_{α_{2}} (f) w_{2} x_{α_{1}} (g) w_{1} B \end{matrix}

Case: $exp (c e_{2 α_{1} + α_{2}}) = x_{2 α_{1} + α_{2}} (c) :$

Cosets starting with $x_{α_{1}} (d) :$

\begin{matrix} x_{2 α_{1} + α_{2}} (c) \cdot x_{α_{1}} (d) w_{1} B & = & x_{α_{1}} (d) x_{2 α_{1} + α_{2}} (c) w_{1} B \\ = & x_{α_{1}} (d) w_{1} x_{α_{2}} (c) B \\ = & x_{α_{1}} (d) w_{1} B \\ x_{2 α_{1} + α_{2}} (c) \cdot x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} B & = & x_{α_{1}} (d) w_{1} x_{α_{2}} (e) x_{α_{2}} (e) w_{2} B \\ = & x_{α_{1}} (d) w_{1} x_{α_{2}} (e + c) w_{2} B \\ x_{2 α_{1} + α_{2}} (c) \cdot x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} B & = & x_{α_{1}} (d) w_{1} x_{α_{2}} (c) x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} B \\ = & x_{α_{1}} (d) w_{1} x_{α_{2}} (e + c) w_{2} x_{α_{1}} (f) w_{1} B \\ x_{2 α_{1} + α_{2}} (c) \cdot x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} x_{α_{2}} (g) w_{2} B & = & x_{α_{1}} (d) w_{1} x_{α_{2}} (c) x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} x_{α_{2}} (g) w_{2} B \\ = & x_{α_{1}} (d) w_{1} x_{α_{2}} (e + c) w_{2} x_{α_{1}} (f) w_{1} x_{α_{2}} (g) w_{2} B \end{matrix}

Cosets starting with $x_{α_{2}} (d) :$

\begin{matrix} x_{2 α_{1} + α_{2}} (c) \cdot x_{α_{2}} (d) w_{2} B & = & x_{α_{2}} (d) w_{2} x_{2 α_{1} + α_{2}} (c) B \\ = & x_{α_{2}} (d) w_{2} B \\ x_{2 α_{1} + α_{2}} (c) \cdot x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} B & = & x_{α_{2}} (d) w_{2} x_{2 α_{1} + α_{2}} (c) x_{α_{1}} (e) w_{1} B \\ = & x_{α_{2}} (d) w_{2} (e) w_{1} x_{α_{2}} (c) B \\ = & x_{α_{2}} (d) w_{2} (e) w_{1} B \\ x_{2 α_{1} + α_{2}} (c) \cdot x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B & = & x_{α_{2}} (d) w_{2} x_{2 α_{1} + α_{2}} (c) x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (c) x_{α_{2}} (f) w_{2} B \\ = & x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f + c) w_{2} B \end{matrix}

$exp (e_{α_{2}} + e_{2 α_{1} + α_{2}}) = x_{α_{2}} (1) x_{2 α_{1} + α_{2}} (1) :$

Cosets starting with $x_{α_{2}} (c) :$

\begin{matrix} x_{α_{2}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{2}} (c) w_{2} B & = & x_{α_{2}} (c + 1) w_{2} x_{2 α_{1} + α_{2}} (1) B \\ = & x_{α_{2}} (c + 1) w_{2} B \\ x_{α_{2}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{2}} (c) w_{2} x_{α_{1}} (d) w_{1} B & = & x_{α_{2}} (c + 1) w_{2} x_{2 α_{1} + α_{2}} (1) x_{α_{1}} (d) w_{1} B \\ = & x_{α_{2}} (c + 1) w_{2} x_{α_{1}} (d) w_{1} x_{α_{2}} (1) B \\ = & x_{α_{2}} (c + 1) w_{2} x_{α_{1}} (d) w_{1} B \\ x_{α_{2}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{2}} (c) w_{2} x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} B & = & x_{α_{2}} (c + 1) w_{2} x_{2 α_{1} + α_{2}} (1) x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} B \\ = & x_{α_{2}} (c + 1) w_{2} x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} B \\ = & x_{α_{2}} (c + 1) w_{2} x_{α_{1}} (d) w_{1} x_{α_{2}} (e + 1) w_{2} B \\ x_{α_{2}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{2}} (c) w_{2} x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} B & = & x_{α_{2}} (c + 1) w_{2} x_{2 α_{1} + α_{2}} (1) x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} B \\ = & x_{α_{2}} (c + 1) w_{2} x_{α_{1}} (d) w_{1} x_{α_{2}} (1) x_{α_{2}} (e) w_{2} x_{α_{1}} (f) w_{1} B \\ = & x_{α_{2}} (c + 1) w_{2} x_{α_{1}} (d) w_{1} x_{α_{2}} (e + 1) w_{2} x_{α_{1}} (f) w_{1} B \end{matrix}

Cosets starting with $x_{α_{1}} (c) :$

\begin{matrix} x_{α_{2}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{1}} (c) w_{1} B & = & x_{α_{2}} (1) \cdot x_{α_{1}} (c) w_{1} B \\ = & x_{α_{1}} (c) w_{1} B \\ x_{α_{2}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{1}} (c) w_{1} x_{α_{2}} (d) w_{2} B & = & x_{α_{2}} (1) \cdot x_{α_{1}} (c) w_{1} x_{α_{2}} (d + 1) w_{2} B \\ = & x_{α_{1}} (c) w_{1} x_{α_{2}} (c + d + 1) w_{2} B \\ x_{α_{2}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{1}} (c) w_{1} x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} B & = & x_{α_{2}} (1) \cdot x_{α_{1}} (c) w_{1} x_{α_{2}} (d + 1) w_{2} x_{α_{1}} (e) w_{1} B \\ = & x_{α_{1}} (c) w_{1} x_{α_{2}} (c + d + 1) w_{2} x_{α_{1}} (c + f) w_{1} B \end{matrix}

Case: $exp (e_{α_{1}} + e_{α_{2}}) = x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) x_{2 α_{1} + α_{2}} (1) :$

Cosets starting with $x_{α_{1}} (c) :$

\begin{matrix} \begin{matrix} x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{1}} (c) w_{1} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) \cdot x_{α_{1}} (c) w_{1} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) \cdot x_{α_{1}} (c + 1) w_{1} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{1}} (c + 1) w_{1} B \\ = & x_{α_{1}} (c + 1) w_{1} B \end{matrix} \\ \begin{matrix} x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{1}} (c) w_{1} x_{α_{2}} (d) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) \cdot x_{α_{1}} (c) w_{1} x_{α_{2}} (d + 1) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) \cdot x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (d + 1) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (c + d + 2) w_{2} B \\ = & x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (d + 1) w_{2} B \end{matrix} \\ \begin{matrix} x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{1}} (c) w_{1} x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) \cdot x_{α_{1}} (c) w_{1} x_{α_{2}} (d + 1) w_{2} x_{α_{1}} (e) w_{1} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) \cdot x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (d + 1) w_{2} x_{α_{1}} (e) w_{1} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (d + c + 2) w_{2} x_{α_{1}} (e + c + 1) w_{1} B \\ = & x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (d + 1) w_{2} x_{α_{1}} (e + c + 3 / 2) w_{1} B \end{matrix} \\ \begin{matrix} x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{1}} (c) w_{1} x_{α_{2}} (d) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) \cdot x_{α_{1}} (c) w_{1} x_{α_{2}} (d + 1) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) \cdot x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (d + 1) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{1}} (c + 1) x_{α_{2}} (1) x_{α_{1} + α_{2}} (- c d) x_{2 α_{1} + α_{2}} (c^{2} d) w_{1} x_{α_{2}} (d + 1) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (1) x_{2 α_{1} + α_{2}} (1) w_{1} x_{α_{1} + α_{2}} (c d) x_{α_{2}} (c^{2} d) x_{α_{2}} (d + 1) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (1) x_{2 α_{1} + α_{2}} (1) w_{1} x_{α_{1} + α_{2}} (c d) x_{α_{2}} (c^{2} d + d - 1) w_{2} x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (c^{2} d + d - 1) w_{2} x_{2 α_{1} + α_{2}} (1) x_{α_{1}} (c d) x_{α_{1}} (e) w_{1} x_{α_{2}} (f) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (c^{2} d + d - 1) w_{2} x_{α_{1}} (e + c d) x_{2 α_{1} + α_{2}} (1) w_{1} x_{α_{2}} (f) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (c^{2} d + d - 1) w_{2} x_{α_{1}} (e + c d) w_{1} x_{α_{2}} (1) x_{α_{2}} (f) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (c^{2} d + d - 1) w_{2} x_{α_{1}} (e + c d) w_{1} x_{α_{2}} (f + 1) w_{2} B \\ = & x_{α_{1}} (c + 1) w_{1} x_{α_{2}} (c^{2} d + d - c - 2) w_{2} x_{α_{1}} (e + c d + 1 / 2) w_{1} x_{α_{2}} (f + 1) w_{2} B \end{matrix} \end{matrix}

Cosets starting with $x_{α_{2}} (d) :$

\begin{matrix} \begin{matrix} x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{2}} (c) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) \cdot x_{α_{2}} (c) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) \cdot x_{α_{2}} (c) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{2}} (c + 1) w_{2} B \\ = & x_{α_{2}} (c + 1) w_{2} B \end{matrix} \\ \begin{matrix} x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{2}} (c) w_{2} x_{α_{1}} (d) w_{1} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) \cdot x_{α_{2}} (c) w_{2} x_{α_{1}} (d) w_{1} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) \cdot x_{α_{2}} (c) w_{2} x_{α_{1}} (c + d) w_{1} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{2}} (c + 1) w_{2} x_{α_{1}} (c + d) w_{1} B \\ = & x_{α_{2}} (c + 1) w_{2} x_{α_{1}} (c + d + 1 / 2) w_{1} B \end{matrix} \\ \begin{matrix} x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) x_{2 α_{1} + α_{2}} (1) \cdot x_{α_{2}} (c) w_{2} x_{α_{1}} (d) w_{1} x_{α_{2}} (e) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) x_{α_{1}} (1) \cdot x_{α_{2}} (c) w_{2} x_{α_{1}} (d) w_{1} x_{α_{2}} (e + 1) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) x_{α_{2}} (1) \cdot x_{α_{2}} (c) w_{2} x_{α_{1}} (c + d) w_{1} x_{α_{2}} (e + 1) w_{2} B \\ = & x_{α_{1} + α_{2}} (1 / 2) \cdot x_{α_{2}} (c + 1) w_{2} x_{α_{1}} (c + d) w_{1} x_{α_{2}} (e + 1) w_{2} B \\ = & x_{α_{2}} (c + 1) w_{2} x_{α_{1}} (c + d + 1 / 2) w_{1} x_{α_{2}} (e + 1) w_{2} B \end{matrix} \end{matrix}

The varieties $ℬ_{(s, n)}$

We examine the varieties $ℬ_{(s, n)},$ which are the cosets in $G / B$ fixed by both $s$ and $exp (n),$ for each pair $(s, n)$ listed in Theorems 3.19-3.23. The Kazhdan-Lusztig classification calls for an examination of the action of the simultaneous centralizer in $G$ of $s$ and $n$ on $ℬ_{s, n} .$ Note that $Z (G) ≅ ℤ_{2}$ is always contained in this centralizer, but acts trivially on $ℬ_{s, n} .$ Thus in the discussion below, we let $C (s, n)$ be the quotient of the centralizer of $s$ and $n$ by the center.

Generic $q :$

If $n = 0,$ then $ℬ_{s, n} = ℬ_{n},$ so we only specify $ℬ_{s, n}$ if $n \neq 0 .$ When $n = 0,$ the homology $H_{*} (ℬ_{s_{t}}, 0)$ is the principal series module $M (t) .$

(s, n) = (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}}),

then $ℬ_{s, n}$ consists of $B,$ $x_{α_{1}} (c) w_{1} B,$ $w_{1} w_{2} B,$ and $w_{1} w_{2} w_{1} B .$ This is a single $ℙ^{1}$ and two points. The centralizer of $s$ is generated by $D,$ $x_{α_{1}} (c),$ and $x_{- α_{1}} (c),$ so that $C_{G} (s, n)$ is generated by $Z (G),$ ${d_{α_{1}} (z) ∣ z \in ℂ^{\times}}$ and $x_{- α_{1}} (c),$ and $C (s, n)$ is trivial.

(s, n) = (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{1} + α_{2}}),

then $ℬ_{s, n}$ consists of $B,$ $x_{α_{1}} (c) w_{1} B,$ $w_{2} B,$ and $w_{1} w_{2} B .$ This is a single $ℙ^{1}$ and two points. The centralizer of $s$ is generated by $D,$ $x_{α_{1}} (c),$ and $x_{- α_{1}} (c),$ so that $C_{G} (s, n)$ is generated by $Z (G),$ ${d_{α_{1}} (z) d_{α_{2}} (z^{- 1}) ∣ z \in ℂ^{\times}}$ and $d_{α_{1}} (- 1) w_{1} .$ Then $C (s, n) = {i d, d_{α_{1}} (- 1) w_{1}} .$

We examine how $C (s, n)$ acts on $H_{•} (ℬ_{s, n}) .$

\begin{matrix} d_{α_{1}} (- 1) w_{1} B = w_{1} B, \\ \begin{matrix} d_{α_{1}} (- 1) w_{1} \cdot x_{α_{1}} (c) w_{1} B & = & d_{α_{1}} (- 1) x_{- α_{1}} (- c) w_{1} w_{1} B \\ = & x_{- α_{1}} (c) H_{α_{1}} (- 1) B \\ = & x_{α_{1}} (c^{- 1}) w_{1} B, \end{matrix} \\ H_{α_{1}} (- 1) w_{1} \cdot w_{2} B = w_{1} w_{2} B, and \\ H_{α_{1}} (- 1) w_{1} \cdot w_{1} w_{2} B = w_{2} B . \end{matrix}

This is a homeomorphism on $ℙ^{1},$ and switches the two points. $H_{0} (ℬ_{s, n}) = ℤ^{3}$ be generated by $α,$ β, and $γ,$ which are the generators of $H_{0} (ℙ^{1}),$ and of the points $H_{0} (w_{2} B)$ and $H_{0} (w_{1} w_{2}) B,$ respectively. Then the component $H_{•} {(ℬ_{s, n})}^{χ}$ is 1-dimensional, generated by $β - γ,$ where $χ$ is the non-trivial representation of $ℤ_{2} .$ $H_{•} {(ℬ_{s, n})}^{triv}$ is 3-dimensional, generated by $β + γ$ and $H_{•} (ℙ^{1}),$ where triv is the trivial representation of $ℤ_{2} .$

\begin{matrix} (s_{t_{1, q^{2}}}, e_{α_{2}}) & (s_{t_{1, q^{2}}}, e_{α_{1} + α_{2}}) \end{matrix}

(s, n) = (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 10 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}], e_{α_{1}}),

then $ℬ_{s, n}$ consists of $B,$ $x_{α_{2}} (c) w_{2} B,$ $w_{2} w_{1} B,$ and $w_{2} w_{1} w_{2} B,$ which is one copy of $ℙ^{1}$ and two points. The centralizer of $s$ is generated by $D,$ $x_{α_{2}} (c),$ and $x_{- α_{2}} (c),$ so that $C_{G} (s, n)$ is generated by $Z (G),$ ${d_{α_{2}} (z) ∣ z \in ℂ^{\times}}$ and $x_{- α_{2}} (c),$ and $C (s, n)$ is trivial.

\begin{matrix} (s_{t_{q^{2}, 1}}, e_{α_{1}}) \end{matrix}

(s, n) = (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}),

then $ℬ_{s, n}$ consists of $B,$ $w_{1} B,$ $x_{α_{2}} (c) w_{2} B,$ and $x_{α_{2}} (c) w_{2} w_{1} B,$ which is two disjoint copies of $ℙ^{1} .$ The centralizer of $s$ is generated by $Z (G),$ $D,$ $x_{α_{2}} (c),$ and $x_{- α_{2}} (c),$ so that $C_{G} (s, n)$ is generated by $Z (G),$ ${d_{α_{1}} (z) d_{α_{2}} (z^{- 2}) ∣ z \in ℂ^{\times}},$ $x_{α_{2}} (c),$ and $x_{- α_{2}} (c),$ and $C (s, n)$ is trivial.

(s, n) = (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & - q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}),

then $ℬ_{s, n}$ consists of $B,$ $w_{1} B,$ $x_{α_{2}} (c) w_{2} B,$ and $x_{α_{2}} (c) w_{2} w_{1} B,$ which is two disjoint copies of $ℙ^{1} .$ The centralizer of $s$ is generated by $D,$ $x_{α_{2}} (c),$ and $x_{- α_{2}} (c),$ so that $C_{G} (s, n)$ is generated by $Z (G),$ ${d_{α_{1}} (z) d_{α_{2}} (z^{- 2}) ∣ z \in ℂ^{\times}}$ and $x_{α_{2}} (c),$ and $C (s, n)$ is trivial.

\begin{matrix} (s_{t_{\pm q, 1}}, e_{2 α_{1} + α_{2}}) \end{matrix}

(s, n) = (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}], e_{α_{1}}),

then $ℬ_{s, n}$ consists of four points - $B,$ $w_{2} B,$ $w_{2} w_{1} B,$ and $w_{2} w_{1} w_{2} B .$ The centralizer of $s$ is $D,$ so that $C_{G} (s, n) = {\pm d_{α_{2}} (z) ∣ z \in ℂ^{\times}}$ and $C (s, n)$ is trivial.

(s, n) = (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}], e_{α_{2}}),

then $ℬ_{s, n}$ consists of four points - $B,$ $w_{1} B,$ $w_{1} w_{2} B,$ and $w_{1} w_{2} w_{1} B .$ The centralizer of $s$ is $D,$ so that $C_{G} (s, n) = {\pm d_{α_{1}} (z) ∣ z \in ℂ^{\times}}$ and $C (s, n)$ is trivial.

(s, n) = (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}], e_{α_{1}} + e_{α_{2}}),

then $ℬ_{s, n} = {B} .$ The centralizer of $s$ is $D,$ so that $C_{G} (s, n) = {\pm d_{α_{1}} (z^{- 1}) d_{α_{2}} (z) ∣ z \in ℂ^{\times}}$ and $C (s, n)$ is trivial.

\begin{matrix} (s_{t_{q^{2}, q^{2}}}, e_{α_{1}}) & (s_{t_{q^{2}, q^{2}}}, e_{α_{2}}) \end{matrix} \begin{matrix} (s_{t_{q^{2}, q^{2}}}, e_{α_{1} + α_{2}}) \end{matrix}

(s, n) = (\pm [\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}}),

(s, n) = ([\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}),

then $ℬ_{s, n}$ consists of four points - $B,$ $w_{1} B,$ $w_{2} B,$ and $w_{2} w_{1} B .$ The centralizer of $s$ is $D,$ so that $C_{G} (s, n) = {\pm d_{α_{1}} (z) d_{α_{2}} (z^{- 2}) ∣ z \in ℂ^{\times}}$ and $C (s, n)$ is trivial.

(s, n) = ([\begin{matrix} - q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & - q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}} + e_{2 α_{1} + α_{2}}),

then $ℬ_{n}$ consists of two points - $B$ and $w_{1} B .$ The centralizer of $s$ is $D,$ so that $C_{G} (s, n) = {\pm 1, \pm d_{α_{1}} (- 1)} ≅ ℤ_{2}^{2} .$ However, $C (s, n) = C_{G} (s, n)$ acts trivially on $ℬ_{s, n},$ so ${(ℬ_{s, n})}^{triv} = ℬ_{n}^{s} .$

\begin{matrix} (s_{t_{- 1, q^{2}}}, e_{α_{2}}) & (s_{t_{- 1, q^{2}}}, e_{2 α_{1} + α_{2}}) \end{matrix} \begin{matrix} (s_{t_{- 1, q^{2}}}, e_{α_{2}} + e_{2 α_{1} + α_{2}}) \end{matrix}

For $z \neq 1, q^{\pm 2}, q^{- 4}, q^{- 6},$ if

(s, n) = (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & q^{2} z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & q^{- 2} z^{- 1 / 2} \end{matrix}], e_{α_{1}}),

For $z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2},$ if

(s, n) = (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], e_{α_{2}}),

\begin{matrix} (s_{t_{q^{2}, z}}, e_{α_{1}}) & (s_{t_{z, q^{2}}}, e_{α_{2}}) \end{matrix}

$q^{8} = 1 (ℓ = 4) :$

The varieties $ℬ_{s, n}$ from the previous section change for the orbit of the semisimple element $[\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}],$ which now includes $[\begin{matrix} q & 0 & 0 & 0 \\ 0 & - q & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & - q^{- 1} \end{matrix}] .$ In this case $𝔤_{q}^{s}$ contains 6 non-zero orbits, 3 of which are not present for generic $q .$

Note that $(s_{t_{q^{- 4}, q^{2}}}, e_{α_{2}}) = w_{2} w_{1} \cdot (s_{t_{q^{2}, q^{2}}}, e_{- 2 α_{1} - α_{2}}) .$ If

(s, n) = ([\begin{matrix} q^{- 3} & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{3} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}}),

then $s = H_{α_{2}} (q^{- 2}) H_{α_{1}} (q) .$ Then $ℬ_{s, n}$ consists of four points - $B,$ $w_{1} B,$ $w_{1} w_{2} B,$ and $w_{1} w_{2} w_{1} B .$ The centralizer of $s$ is $D,$ so that $C_{G} (s, n)$ is ${\pm d_{α_{1}} (z) ∣ z \in ℂ^{\times}}$ and $C (s, n)$ is trivial.

Next, $(s_{t_{q^{- 2}, q^{- 2}}}, e_{α_{2}} + e_{2 α_{1} + α_{2}}) = w_{2} w_{1} \cdot (s_{t_{q^{2}, q^{2}}}, e_{α_{2}} + e_{- 2 α_{1} - α_{2}}) .$

(s, n) = ([\begin{matrix} q^{- 3} & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{3} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{2}} + e_{2 α_{1} + α_{2}}),

then $ℬ_{s, n}$ consists of $B$ and $w_{1} B .$ The centralizer of $s$ is $D,$ so that $C_{G} (s, n) = {1 \pm, \pm d_{α_{1}} (- 1)}$ and $C (s, n)$ is $ℤ_{2} .$ However, $C (s, n)$ acts trivially on $ℬ_{s, n},$ so that ${(ℬ_{s, n})}^{triv} = ℬ_{s, n} .$

Finally, $(s_{t_{q^{2}, q^{2}, 1}}, e_{α_{1}} + e_{α_{2}}) = w_{2} w_{1} w_{2} \cdot (s_{t_{q^{2}, q^{2}, 0}}, e_{α_{1}} + e_{- 2 α_{1} - α_{2}} .)$ If

(s, n) = ([\begin{matrix} q^{- 3} & 0 & 0 & 0 \\ 0 & q^{- 1} & 0 & 0 \\ 0 & 0 & q^{3} & 0 \\ 0 & 0 & 0 & q \end{matrix}], e_{α_{1}} + e_{α_{2}}),

then $s = H_{α_{1}} (q^{- 1}) H_{α_{2}} (q^{- 4}) .$ Then $ℬ_{s, n}$ consists of $B .$ The centralizer of $s$ is $D$ so that $C (s, n)$ is trivial.

\begin{matrix} (s_{t_{q^{2}, q^{2}}}, e_{- 2 α_{1} - α_{2}}) & (s_{t_{q^{2}, q^{2}}}, e_{α_{2}} + e_{- 2 α_{1} - α_{2}}) \end{matrix} \begin{matrix} (s_{t_{q^{2}, q^{2}}}, e_{α_{1}} + e_{- 2 α_{1} - α_{2}}) \end{matrix}

$q^{6} = 1 (ℓ = 3) :$

The varieties $ℬ_{s, n}$ change from the generic case for $s$ in the orbit of $[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}],$ which is now in the same orbit as $\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{3} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 3} \end{matrix}],$ depending on whether $q^{3}$ is 1 or $- 1 .$ The $C_{G} (s_{t_{q^{2}, 1}}) -orbits$ in $𝒩_{q}^{s_{t_{q^{2}, 1}}}$ are represented by $e_{α_{1}},$ $e_{- 2 α_{1} - α_{2}},$ and $e_{α_{1}} + e_{- 2 α_{1} - α_{2}} .$

(s, n) = (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}], e_{α_{1}}),

then $ℬ_{s, n}$ consists of $B,$ $x_{α_{2}} (c) w_{2} B,$ $w_{2} w_{1} B,$ and $w_{2} w_{1} w_{2} B,$ which is one copy of $ℙ^{1}$ and two points. The centralizer of $s$ is generated by $x_{\pm α_{2}} (c)$ and $D,$ so that $C_{G} (s, n)$ is generated by ${x_{- α_{2}} (c) ∣ c \in ℂ}$ and ${\pm d_{α_{2}} (z) ∣ z \in ℂ^{\times}} .$ Then $C (s, n)$ is trivial.

Note that $(s_{t_{q^{- 2}, q^{- 2}}}, e_{α_{2}}) = w_{1} w_{2} \cdot (s_{t_{q^{2}, 1}}, e_{- 2 α_{1} - α_{2}}) .$

(s, n) = (\pm [\begin{matrix} q^{2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & q^{- 2} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], e_{α_{2}}),

then $ℬ_{s, n}$ consists of $B,$ $w_{1} B,$ $w_{1} x_{α_{2}} (c) w_{2} B,$ and $w_{1} x_{α_{2}} (c) w_{2} w_{1} B,$ which is two copies of $ℙ^{1} .$ The centralizer of $s$ is generated by $x_{\pm (2 α_{1} + α_{2})} (c)$ and $D,$ so that $C_{G} (s, n)$ is generated by $x_{\pm (2 α_{1} + α_{2})} (c)$ for $c \in ℂ$ and ${\pm d_{α_{1}} (z) ∣ z \in ℂ^{\times}} .$ Then $C (s, n)$ is trivial.

Finally, $(s_{t_{q^{2}, q^{2}}}, e_{α_{1}} + e_{α_{2}}) = w_{2} w_{1} w_{2} \cdot (s_{t_{q^{2}, 1}}, e_{α_{1}} + e_{- 2 α_{1} - α_{2}}) .$

(s, n) = (\pm [\begin{matrix} q^{2} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & q^{- 2} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}], e_{α_{2}} + e_{α_{2}}),

then $ℬ_{s, n}$ consists of only $B .$ The centralizer of $s$ is generated by $x_{\pm (2 α_{1} + α_{2})} (c)$ and $D,$ so that $C_{G} (s, n)$ is generated by $x_{2 α_{1} + α_{2}} (c)$ for $c \in ℂ,$ and ${\pm d_{α_{1}} (z) d_{α_{2}} (z^{- 1}) ∣ z \in ℂ^{\times}} .$ Hence $C (s, n)$ is trivial.

\begin{matrix} (s_{t_{q^{2}, 1}}, e_{α_{1}}) & (s_{t_{q^{2}, 1}}, e_{- 2 α_{1} - α_{2}}) \end{matrix} \begin{matrix} (s_{t_{q^{2}, 1}}, e_{α_{1}} + e_{- 2 α_{1} - α_{2}}) \end{matrix}

$q^{4} = 1 (ℓ = 2) :$

Again, we note only the changes from the generic case. Also note that $ℬ_{s, n} = ℬ_{- s, n} = ℬ_{q^{2} s, n} .$

$t_{1, q^{2}}$

If $s = s_{t_{1, q^{2}}},$ then the $C_{G} (s) -orbits$ in $𝒩_{q}^{s}$ are represented by $e_{\pm α_{2}},$ $e_{\pm (α_{1} + α_{2}),}$ $e_{α_{2}} + e_{- α_{1} - α_{2}},$ $e_{α_{2}} + e_{- 2 α_{1} - α_{2}},$ and $e_{α_{1} + α_{2}} + e_{- α_{2}} .$

We can check which nilpotent elements satisfy condition 3.1 using the following procedure. First, we note that an element of $𝔤_{q}^{s}$ takes the form

z = a e_{α_{2}} + b e_{α_{1} + α_{2}} + c e_{2 α_{1} + α_{2}} + d e_{- α_{2}} + f e_{- α_{1} - α_{2}} + g e_{- 2 α_{1} - α_{2}} .

Then for each $n,$ we find an ${𝔰 𝔩}_{2} -triple$ $(n, y, h) .$ By computing the commutator $[y, z],$ we can determine necessary and sufficient conditions for $z$ to commute with $y .$ We then check, using the standard representation of $𝔤,$ whether an element $z$ satisfying those conditions is necessarily nilpotent. We outline this check below.

\begin{matrix} n & y & Condition on Z_{𝔤} (y) & Z_{𝔤} (y) \cap 𝔤_{q}^{s} \subseteq 𝒩 \\ e_{α_{2}} & e_{- α_{2}} & a = b = 0 & no \\ e_{- α_{2}} & e_{- α_{2}} & d = f = 0 & no \\ e_{α_{1} + α_{2}} & e_{- α_{1} - α_{2}} & a = b = c = 0 & yes \\ e_{- α_{1} - α_{2}} & e_{α_{1} + α_{2}} & d = f = g = 0 & yes \\ e_{α_{2}} + e_{- α_{1} - α_{2}} & 4 e_{- α_{2}} + 3 e_{α_{1} + α_{2}} & a = f = g = 0 = 4 b - 3 d & yes \\ e_{- α_{2}} + e_{α_{1} + α_{2}} & 4 e_{α_{2}} + 3 e_{- α_{1} - α_{2}} & b = c = d = 0 = 4 f - 3 a & yes \\ e_{α_{2}} + e_{- 2 α_{1} - α_{2}} & e_{- α_{2}} + e_{2 α_{1} + α_{2}} & a = g = 0 = b - f & no \end{matrix}

Specifically, $e_{2 α_{1} + α_{2}} + f_{2 α_{1} + α_{2}}$ commutes with $e_{\pm α_{2}},$ and $e_{α_{1} + α_{2}} - e_{- α_{1} - α_{2}}$ commutes with $e_{- α_{2}} + e_{2 α_{1} + α_{2}},$ so that $e_{\pm α_{2}}$ and $e_{α_{2}} + e_{- 2 α_{1} - α_{2}}$ are not considered.

(s, n) = ([\begin{matrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{1} + α_{2}}),

then $ℬ_{s, n}$ consists of $B,$ $x_{α_{1}} (c) w_{1} B,$ $w_{2} B,$ and $w_{1} w_{2} B,$ a $ℙ^{1}$ and two points. The centralizer of $s$ is generated by $D$ and $x_{\pm α_{1}} (c),$ so that $C_{G} (s, n)$ is generated by ${d_{α_{1}} (z) d_{α_{2}} (z^{- 1}) ∣ z \in ℂ^{\times}}$ and $d_{α_{1}} (- 1) w_{1} .$ Then

C (s, n) = {1, d_{α_{1}} (- 1) w_{1}} ≅ ℤ_{2} .

(s, n) = w_{0} \cdot (s_{t_{1, q^{2}}}, e_{- α_{1} - α_{2}}) = ([\begin{matrix} q^{- 1} & 0 & 0 & 0 \\ 0 & q^{- 1} & 0 & 0 \\ 0 & 0 & q & 0 \\ 0 & 0 & 0 & q \end{matrix}], e_{α_{1} + α_{2}}),

then $ℬ_{s, n}$ consists of $B,$ $x_{α_{1}} (c) w_{1} B,$ $w_{2} B,$ and $w_{1} w_{2} B,$ a $ℙ^{1}$ and two points. The centralizer of $s$ is generated by $D$ and $x_{\pm α_{1}} (c),$ so that $C_{G} (s, n)$ is generated by ${d_{α_{1}} (z) d_{α_{2}} (z^{- 1}) ∣ c \in ℂ^{\times}}$ and $d_{α_{1}} (- 1) w_{1} .$ Then $C (s, n) = {1, d_{α_{1}} (- 1) w_{1}} ≅ ℤ_{2} .$

In both of these cases, as in the case of generic $q,$ the action of $C (s, n)$ on $ℬ_{s, n}$ switches the points $w_{2} B$ and $w_{1} w_{2} B,$ and is a homeomorphism on $ℙ^{1} = {x_{α_{1}} (c) w_{1} B ∣ c \in ℂ} \cup {B} .$ Thus the component of $H_{•} (ℬ_{s, n})$ corresponding to the sign representation is 1-dimensional, spanned by $[w_{2} B] + [w_{1} w_{2} B],$ while the component corresponding to the trivial representation is 3-dimensional.

(s, n) = w_{1} w_{2} w_{1} \cdot (s_{t_{1, q^{2}}}, e_{α_{2}} + e_{- α_{1} - α_{2}}) = ([\begin{matrix} q & 0 & 0 & 0 \\ 0 & q^{- 1} & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & q \end{matrix}], e_{α_{2}} + e_{α_{1}}),

then $ℬ = {B} .$ The centralizer of $s$ is generated by $D$ and $x_{\pm α_{1}} (c),$ so that $C (s, n)$ is trivial.

(s, n) = w_{2} \cdot (s_{t_{1, q^{2}}}, e_{α_{1} + α_{2}} + e_{- α_{2}}) = ([\begin{matrix} q^{- 1} & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & q & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{α_{1}} + e_{α_{2}}),

then $ℬ_{s, n} = {B} .$ The centralizer of $s$ is generated by $D$ and $x_{α_{1}} (c),$ so that $C (s, n)$ is trivial.

\begin{matrix} (s_{t_{1, q^{2}}}, e_{α_{1} + α_{2}}) & (s_{t_{1, q^{2}}}, e_{- α_{1} - α_{2}}) \\ (s_{t_{1, q^{2}}}, e_{α_{2}} + e_{- α_{1} - α_{2}}) & (s_{t_{1, q^{2}}}, e_{α_{1} + α_{2}} + e_{- α_{2}}) \end{matrix}

Thus, the triples $(s_{s_{2} t}, e_{α_{1}} + e_{α_{2}}, 1)$ and $(s_{s_{1} s_{2} s_{1} t}, e_{α_{1}} + e_{α_{2}}, 1)$ must correspond to the 1-dimensional modules with weights $s_{2} t$ and $s_{1} s_{2} s_{1} t,$ respectively, since the homology of those varieties are each 1-dimensional. Then $(s_{t}, e_{α_{1} + α_{2}}, 1)$ and $(s_{w_{0} t}, e_{α_{1} + α_{2}}, 1)$ correspond to the 2-dimensional modules, which have weights $t$ and $w_{0} t .$ However, this leaves us with two triples not yet assigned to a module, $(s_{t}, e_{α_{1} + α_{2}}, - 1)$ and $(s_{t}, e_{α_{1} + α_{2}}, - 1),$ and every module has been accounted for. This is the first case where Grojnowski’s condition 3.2 seems to be missing some information about the representation $χ$ in the triple $(s, n, χ) .$ In particular, eliminating the triples $(s_{t}, e_{α_{1} + α_{2}}, - 1)$ and $(s_{t}, e_{α_{1} + α_{2}}, - 1)$ from our indexing set makes the correspondence a bijection.

$t_{q^{2}, 1}$

(s, n) = ([\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{2} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 2} \end{matrix}], e_{α_{1}}),

then $ℬ_{s, n}$ consists of $B,$ $x_{α_{2}} (c) w_{2} B,$ $w_{2} w_{1} B,$ and $w_{2} w_{1} x_{α_{2}} (c) w_{2} B,$ two disjoint copies of $ℙ^{1} .$ The centralizer of $s$ is generated by $x_{\pm α_{2}} (c)$ and $D,$ so that $C_{G} (s, n)$ is generated by ${x_{- α_{2}} (c) ∣ c \in ℂ}$ and ${\pm d_{α_{2}} (z) ∣ z \in ℂ^{\times}} .$ Then $C (s, n)$ is trivial.

\begin{matrix} (s_{t_{q^{2}, 1}}, e_{α_{1}}) \end{matrix}

$t_{q, 1}$

(s, n) = (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q^{- 1} \end{matrix}], e_{2 α_{1} + α_{2}}),

then $ℬ_{s, n}$ consists of $B,$ $w_{1} B,$ $x_{α_{2}} (c) w_{2} B,$ and $x_{α_{2}} (c) w_{2} w_{1} B,$ two disjoint copies of $ℙ^{1} .$ The centralizer of $s$ is generated by $x_{\pm α_{2}} (c)$ and $D,$ so that $C_{G} (s, n)$ is generated by $Z (G),$ ${x_{- α_{2}} (c) ∣ c \in ℂ}$ and $d_{α_{1}} (- 1) .$ The group $C (s, n)$ is isomorphic to $ℤ_{2},$ generated by $d_{α_{1}} (- 1),$ but this group acts trivially on $ℬ_{s, n} .$

(s, n) = w_{0} \cdot (s_{t_{q, 1}}, e_{- 2 α_{1} - α_{2}}) = (\pm [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & q^{- 1} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & q \end{matrix}], e_{2 α_{1} + α_{2}}),

then $ℬ_{s, n}$ consists of $B,$ $w_{1} B,$ $x_{α_{2}} (c) B,$ and $x_{α_{2}} (c) w_{2} w_{1} B,$ two disjoint copies of $ℙ^{1} .$ The centralizer of $s$ is generated by $x_{\pm α_{2}} (c)$ and $D,$ so that $C_{G} (s, n)$ is generated by $Z (G),$ ${x_{- α_{2}} (c) ∣ c \in ℂ}$ and $d_{α_{1}} (- 1) .$ The group $C (s, n)$ is isomorphic to $ℤ_{2},$ generated by $d_{α_{1}} (- 1),$ but this group acts trivially on $ℬ_{s, n} .$

\begin{matrix} (s_{t_{\pm q, 1}}, e_{- 2 α_{1} - α_{2}}) & (s_{t_{\pm q, 1}}, e_{2 α_{1} + α_{2}}) \end{matrix}

If $z \neq 1, q^{\pm 2}, q^{- 4}, q^{- 6}$ and

(s, n) = (\pm [\begin{matrix} z^{1 / 2} & 0 & 0 & 0 \\ 0 & q^{2} z^{1 / 2} & 0 & 0 \\ 0 & 0 & z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & q^{- 2} z^{- 1 / 2} \end{matrix}], e_{α_{1}}),

then $ℬ_{s, n}$ consists of four points - $B,$ $w_{2} B,$ $w_{2} w_{1} B,$ and $w_{2} w_{1} w_{2} B .$ The centralizer of $s$ is $D$ so that $C_{G} (s, n)$ is ${\pm d_{α_{2}} (z) ∣ z \in ℂ^{\times}} .$ Then $C (s, n)$ is trivial.

If $z \neq 1, q^{\pm 2}, q^{- 4}, q^{- 6}$ and

(s, n) = (\pm [\begin{matrix} q^{2} z^{1 / 2} & 0 & 0 & 0 \\ 0 & z^{1 / 2} & 0 & 0 \\ 0 & 0 & q^{- 2} z^{- 1 / 2} & 0 \\ 0 & 0 & 0 & z^{- 1 / 2} \end{matrix}], e_{α_{1}}),

\begin{matrix} (s_{t_{q^{2}, z}}, e_{α_{1}}) & (s_{t_{q^{2}, z}}, e_{- α_{1}}) \end{matrix}

If $z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2}$ and

(s, n) = (\pm [\begin{matrix} q & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], e_{α_{2}}),

then $ℬ_{s, n}$ consists of four points - $B,$ $w_{1} B,$ $w_{1} w_{2} B,$ and $w_{1} w_{2} w_{1} B .$ The centralizer of $s$ is $D$ so that $C_{G} (s, n)$ is ${\pm d_{α_{1}} (z) ∣ z \in ℂ^{\times}} .$ Then $C (s, n)$ is trivial.

If $z \neq \pm 1, q^{\pm 2}, q^{- 4}, - q^{- 2}$ and

(s, n) = (\pm [\begin{matrix} q^{- 1} & 0 & 0 & 0 \\ 0 & z q & 0 & 0 \\ 0 & 0 & q^{- 1} & 0 \\ 0 & 0 & 0 & z^{- 1} q^{- 1} \end{matrix}], e_{α_{2}}),

then $ℬ_{s, n}$ consists of four points - $B,$ $w_{1} B,$ $w_{1} w_{2} B,$ and $w_{1} W_{2} w_{1} B .$ The centralizer of $s$ is $D$ so that $C_{G} (s, n)$ is ${\pm d_{α_{1}} (z) ∣ z \in ℂ^{\times}} .$ Then $C (s, n)$ is trivial.

\begin{matrix} (s_{t_{z, q^{2}}}, e_{α_{2}}) & (s_{t_{z, q^{2}}}, e_{- α_{2}}) \end{matrix}

$q^{2} = 1 (ℓ = 1) :$

When $q^{2} = 1,$ $Z (t) = P (t)$ for any $t,$ and $C_{G} (s)$ is the Lie group generated by $D$ and ${x_{α} (c) ∣ α \in 𝔤_{q}^{s}, c \in ℂ} .$ In fact, if $e_{α}$ and $e_{β}$ are elements of $𝔤_{q}^{s},$ then $e_{α + β} \in 𝔤_{q}^{s}$ as well. Then $𝔤_{q}^{s}$ is a Lie subalgebra of $𝔤$ and $C_{G} (s)$ is its associated Lie Group. Then the $C_{G} (s)$ orbits of $𝒩_{q}^{s}$ are exactly the (adjoint) nilpotent orbits of $𝔤_{q}^{s} .$ In addition, the set of Borel subgroups of $C_{G} (s)$ is precisely $ℬ_{s},$ and the Weyl group of $C_{G} (s)$ is $W_{t},$ the stabilizer of $t$ in $W_{0} .$

Then, the Springer correspondence gives a bijection between irreducible representations of $W_{t}$ and $C_{G} (s_{t}) -orbits$ of pairs $(n, χ),$ where $n$ is a nilpotent element of $𝔤$ and $χ$ is a simple representation of the component group of $C_{G} (s_{t}, n)$ that appears in $H (ℬ_{s, n}) .$ But these are exactly the $G -orbits$ of triples $(s_{t}, n, χ)$ where $χ$ is a simple representation of $C (s, n)$ that appears in $H_{•} (ℬ_{s, n}) .$ Then the orbits of such triples are in bijection with the irreducible representations of $W_{t} .$ In turn, the results of section 1.2.9 show that the irreducible representations of $W_{t}$ are in bijection with the irreducible representations of $\tilde{H}$ with central character $t .$ Thus, if $q^{2} = 1,$ using the Springer correspondences for all the potential groups $W_{t}$ gives a geometric indexing of the irreducible representations of $\tilde{H} .$

Bijections

We summarize the bijections between irreducible representations of $\tilde{H}$ and orbits in

{(s, n) ∣ s \in G^{ss}, n \in 𝔤_{q}^{s}}

paired with representations of $C {(s, n)}^{\circ}$ appearing in $H_{•} (ℬ_{s, n}) .$

Generic $q$

\begin{matrix} \begin{matrix} Central Character & Dimension & Indexing & Weights \\ t_{1, 1} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{- 1, 1} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{1, z} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{1, q^{2}} & 1 & (s_{t}, e_{α_{1} + α_{2}}, - 1) & s_{2} t \\ t_{1, q^{2}} & 1 & (s_{t}, e_{α_{2}}, 1) & w_{0} t \\ t_{1, q^{2}} & 3 & (s_{t}, e_{α_{1} + α_{2}}, 1) & t, s_{2} t \\ t_{1, q^{2}} & 3 & (s_{t}, 0, 1) & w_{0} t, s_{2} w_{0} t \\ t_{q^{2}, 1} & 4 & (s_{t}, 0, 1) & s_{1} t, s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{q^{2}, 1} & 4 & (s_{t}, e_{α_{1}}, 1) & t, s_{1} t, s_{2} s_{1} t \\ t_{q, 1} & 4 & (s_{t}, 0, 1) & s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{q, 1} & 4 & (s_{t}, e_{2 α_{1} + α_{2}}, 1) & t, s_{1} t \\ t_{- q, 1} & 4 & (s_{t}, 0, 1) & s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{- q, 1} & 4 & (s_{t}, e_{2 α_{1} + α_{2}}, 1) & t, s_{1} t \\ t_{z, 1} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{q^{2}, q^{2}} & 1 & (s_{t}, 0, 1) & W_{0} t \\ t_{q^{2}, q^{2}} & 1 & (s_{t}, e_{α_{1}} + e_{α_{2}}, 1) & t \\ t_{q^{2}, q^{2}} & 3 & (s_{t}, e_{α_{1}}, 1) & s_{2} t, s_{1} s_{2} t, s_{2} s_{1} s_{2} t \\ t_{q^{2}, q^{2}} & 3 & (s_{t}, e_{α_{2}}, 1) & s_{1} t, s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{q^{2}, z} & 4 & (s_{t}, 0, 1) & s_{1} t, s_{2} s_{1} t, s_{1} s_{2} s_{1} t, w_{0} t \\ t_{q^{2}, z} & 4 & (s_{t}, e_{α_{1}}, 1) & t, s_{2} t, s_{1} s_{2} t, s_{2} s_{1} s_{2} t \\ t_{- 1, q^{2}} & 2 & (s_{t}, 0, 1) & s_{2} s_{1} s_{2} t, w_{0} t \\ t_{- 1, q^{2}} & 2 & (s_{t}, e_{α_{2}}, 1) & s_{2} t, s_{1} s_{2} t \\ t_{- 1, q^{2}} & 2 & (s_{t}, e_{2 α_{1} + α_{2}}, 1) & s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{- 1, q^{2}} & 2 & (s_{t}, e_{α_{2}} + e_{2 α_{1} + α_{2}}, 1) & t, s_{1} t \\ t_{z, q^{2}} & 4 & (s_{t}, 0, 1) & s_{2} t, s_{1} s_{2} t, s_{2} s_{1} s_{2} t, w_{0} t \\ t_{z, q^{2}} & 4 & (s_{t}, e_{α_{2}}, 1) & t, s_{1} t, s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{z, w} & 8 & (s_{t}, 0, 1) & W_{0} t \end{matrix} \\ Table 27: Geometric Indexing in Type C_{2}, with generic q . \end{matrix}

$q^{8} = 1$

\begin{matrix} \begin{matrix} Central Character & Dimension & Indexing & Weights \\ t_{1, 1} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{- 1, 1} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{1, z} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{1, q^{2}} & 1 & (s_{t}, e_{α_{1} + α_{2}}, - 1) & s_{2} t \\ t_{1, q^{2}} & 1 & (s_{t}, e_{α_{2}}, 1) & w_{0} t \\ t_{1, q^{2}} & 3 & (s_{t}, e_{α_{1} + α_{2}}, 1) & t, s_{2} t \\ t_{1, q^{2}} & 3 & (s_{t}, 0, 1) & s_{1} s_{2} t, s_{2} s_{1} s_{2} t \\ t_{q^{2}, 1} & 4 & (s_{t}, 0, 1) & s_{1} t, s_{2} s_{1} t, s_{1} s_{2} s_{1} t, w_{0} t \\ t_{q^{2}, 1} & 4 & (s_{t}, e_{α_{1}}, 1) & t, s_{2} t, s_{1} s_{2} t, s_{2} s_{1} s_{2} t \\ t_{q, 1} & 4 & (s_{t}, 0, 1) & s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{q, 1} & 4 & (s_{t}, e_{2 α_{1} + α_{2}}, 1) & t, s_{1} t \\ t_{- q, 1} & 4 & (s_{t}, 0, 1) & s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{- q, 1} & 4 & (s_{t}, e_{2 α_{1} + α_{2}}, 1) & t, s_{1} t \\ t_{z, 1} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{q^{2}, q^{2}} & 1 & (s_{s_{1} s_{2} t}, e_{α_{2}}, 1) & W_{0} t \\ t_{q^{2}, q^{2}} & 2 & (s_{s_{1} s_{2} t}, e_{α_{2}} + e_{2 α_{1} + α_{2}}, 1) & s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{q^{2}, q^{2}} & 1 & (s_{s_{2} s_{1} s_{2} t}, e_{α_{1}} + e_{α_{2}}, 1) & s_{2} s_{1} s_{2} t \\ t_{q^{2}, q^{2}} & 1 & (s_{t}, e_{α_{1}} + e_{α_{2}}, 1) & t \\ t_{q^{2}, q^{2}} & 2 & (s_{t}, e_{α_{1}}, 1) & s_{2} t, s_{1} s_{2} t \\ t_{q^{2}, q^{2}} & 1 & (s_{t}, e_{α_{2}}, 1) & s_{1} t \\ t_{q^{2}, z} & 4 & (s_{t}, 0, 1) & s_{1} t, s_{2} s_{1} t, s_{1} s_{2} s_{1} t, w_{0} t \\ t_{q^{2}, z} & 4 & (s_{t}, e_{α_{1}}, 1) & t, s_{2} t, s_{1} s_{2} t, s_{2} s_{1} s_{2} t \\ t_{z, q^{2}} & 4 & (s_{t}, 0, 1) & s_{2} t, s_{1} s_{2} t, s_{2} s_{1} s_{2} t, w_{0} t \\ t_{z, q^{2}} & 4 & (s_{t}, e_{α_{2}}, 1) & t, s_{1} t, s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{z, w} & 8 & (s_{t}, 0, 1) & W_{0} t \end{matrix} \\ Table 28: Geometric Indexing in Type C_{2}, with q^{8} = 1 . \end{matrix}

Note that $(s_{t_{q^{2}, q^{2}}}, 0)$ is omitted from this bijection since $Z_{𝔤} (0) \cap 𝔤_{q}^{s}$ is not contained in $𝒩,$ and thus condition 3.1 omits it.

$q^{6} = 1$

\begin{matrix} \begin{matrix} Central Character & Dimension & Indexing & Weights \\ t_{1, 1} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{- 1, 1} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{1, z} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{1, q^{2}} & 1 & (s_{t}, e_{α_{1} + α_{2}}, - 1) & s_{2} t \\ t_{1, q^{2}} & 1 & (s_{t}, e_{α_{2}}, 1) & s_{1} s_{2} t \\ t_{1, q^{2}} & 3 & (s_{t}, e_{α_{1} + α_{2}}, 1) & t, s_{2} t \\ t_{1, q^{2}} & 3 & (s_{t}, 0, 1) & s_{1} s_{2} t, s_{2} s_{1} s_{2} t \\ t_{q^{2}, 1} & 3 & (s_{t}, 0, 1) & s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{q^{2}, 1} & 3 & (s_{t}, e_{α_{1}}, 1) & t, s_{1} t \\ t_{q^{2}, 1} & 1 & (s_{s_{2} s_{1} t}, e_{α_{2}}, 1) & s_{1} t \\ t_{q^{2}, 1} & 1 & (s_{s_{2} s_{1} t}, e_{α_{1}} + e_{α_{2}}, 1) & s_{2} s_{1} t \\ t_{q, 1} & 4 & (s_{t}, 0, 1) & s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{q, 1} & 4 & (s_{t}, e_{2 α_{1} + α_{2}}, 1) & t, s_{1} t \\ t_{z, 1} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{q^{2}, z} & 4 & (s_{t}, 0, 1) & s_{1} t, s_{2} s_{1} t, s_{1} s_{2} s_{1} t, s_{2} s_{1} s_{2} s_{1} t \\ t_{q^{2}, z} & 4 & (s_{t}, e_{α_{1}}, 1) & t, s_{2} t, s_{1} s_{2} t, s_{2} s_{1} s_{2} t \\ t_{- 1, q^{2}} & 2 & (s_{t}, 0, 1) & s_{2} s_{1} s_{2} t, s_{1} s_{2} s_{1} s_{2} t \\ t_{- 1, q^{2}} & 2 & (s_{t}, e_{α_{2}}, 1) & s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{- 1, q^{2}} & 2 & (s_{t}, e_{2 α_{1} + α_{2}}, 1) & s_{2} t, s_{1} s_{2} t \\ t_{- 1, q^{2}} & 2 & (s_{t}, e_{α_{2}} + e_{2 α_{1} + α_{2}}, 1) & t, s_{1} t \\ t_{z, q^{2}} & 4 & (s_{t}, 0, 1) & s_{2} t, s_{1} s_{2} t, s_{2} s_{1} s_{2} t, s_{1} s_{2} s_{1} s_{2} t \\ t_{z, q^{2}} & 4 & (s_{t}, e_{α_{2}}, 1) & t, s_{1} t, s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{z, w} & 8 & (s_{t}, 0, 1) & W_{0} t \end{matrix} \\ Table 29: Geometric Indexing in Type C_{2}, with q^{6} = 1 . \end{matrix}

$q^{4} = 1$

\begin{matrix} \begin{matrix} Central Character & Dimension & Indexing & Weights \\ t_{1, 1} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{- 1, 1} & 4 & (s_{t}, e_{α_{1}}, 1) & t, s_{1} t \\ t_{1, z} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{1, q^{2}} & 1 & (s_{t}, e_{- α_{2}} + e_{α_{1} + α_{2}}, 1) & s_{2} t \\ t_{1, q^{2}} & 1 & (s_{t}, e_{α_{2}} + e_{- α_{1} + α_{2}}, 1) & s_{1} s_{2} t \\ t_{1, q^{2}} & 2 & (s_{t}, e_{α_{1} + α_{2}}, 1) & t \\ t_{1, q^{2}} & ? & (s_{t}, e_{α_{1} + α_{2}}, 1) & ? \\ t_{1, q^{2}} & 2 & (s_{w_{0} t}, e_{α_{1} + α_{2}}, 1) & s_{2} s_{1} s_{2} t \\ t_{1, q^{2}} & ? & (s_{w_{0} t}, e_{α_{1} + α_{2}}, - 1) & ? \\ t_{q, 1} & 4 & (s_{t}, e_{2 α_{1} + α_{2}}, 1) & t, s_{1} t \\ t_{q, 1} & 4 & (s_{t}, e_{- 2 α_{1} - α_{2}}, 1) & s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{z, 1} & 8 & (s_{t}, 0, 1) & W_{0} t \\ t_{q^{2}, z} & 4 & (s_{t}, e_{α_{1}}, 1) & t, s_{2} t, s_{1} s_{2} t, s_{2} s_{1} s_{2} t \\ t_{q^{2}, z} & 4 & (s_{t}, e_{- α_{1}}, 1) & s_{1} t, s_{2} s_{1} t, s_{1} s_{2} s_{1} t, s_{2} s_{1} s_{2} s_{1} t \\ t_{z, q^{2}} & 4 & (s_{t}, e_{α_{2}}, 1) & t, s_{1} t, s_{2} s_{1} t, s_{1} s_{2} s_{1} t \\ t_{z, q^{2}} & 4 & (s_{t}, e_{- α_{2}}, 1) & s_{2} t, s_{1} s_{2} t, s_{2} s_{1} s_{2} t, s_{1} s_{2} s_{1} s_{2} t \\ t_{z, w} & 8 & (s_{t}, 0, 1) & W_{0} t \end{matrix} \\ Table 30: Geometric Indexing in Type C_{2}, with q^{2} = - 1 . \end{matrix}

For the central character $t_{1, q^{2}}$ when $q^{4} = 1,$ we note that condition 3.1 omits the nilpotent $C_{G} (s_{t}) -orbits$ of 0, $e_{\pm α_{2}},$ and $e_{α_{2}} + e_{- 2 α_{1} - α_{2}}$ from conisderation.

$q^{2} = 1$

\begin{matrix} \begin{matrix} Central Character & Dimension & Indexing & Weights \\ t_{1, 1} & 1 & (s_{t}, 0, 1) & sign \\ t_{1, 1} & 2 & (s_{t}, e_{α_{1}}, 1) \\ t_{1, 1} & 1 & (s_{t}, e_{α_{1}}, - 1) \\ t_{1, 1} & 1 & (s_{t}, e_{α_{2}}, 1) \\ t_{1, 1} & 1 & (s_{t}, e_{α_{1}} + e_{α_{2}}, 1) & triv \\ t_{1, z} & 4 & (s_{t}, 0, 1) & sign \otimes_{{\tilde{H}}_{{1}}} \tilde{H} \\ t_{1, z} & 4 & (s_{t}, e_{α_{1}}, 1) & triv \otimes_{{\tilde{H}}_{{1}}} \tilde{H} \\ t_{q, 1} & 4 & (s_{t}, 0, 1) & sign \otimes_{{\tilde{H}}_{{1}}} \tilde{H} \\ t_{q, 1} & 4 & (s_{t}, e_{2 α_{1} + α_{2}}, 1) & triv \otimes_{{\tilde{H}}_{{1}}} \tilde{H} \\ t_{z, 1} & 4 & (s_{t}, 0, 1) & sign \otimes_{{\tilde{H}}_{{1}}} \tilde{H} \\ t_{z, 1} & 4 & (s_{t}, e_{α_{2}}, 1) & triv \otimes_{{\tilde{H}}_{{1}}} \tilde{H} \\ t_{z, w} & 8 & (s_{t}, 0, 1) & triv \otimes_{ℂ [X]} \tilde{H} \end{matrix} \\ Table 31: Geometric Indexing in Type C_{2}, with q = - 1 . \end{matrix}

Notes and References

This is an excerpt from Matt Davis' Ph.D Thesis entitled Representations of Rank Two Affine Hecke Algebras at Roots of Unity, University of Wisconsin, 2010.

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