Hecke algebras

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 14 May 2010

General Hecke algebras

Let $B$ be an algebra. The subspace $p B p \subseteq B$ is closed under multiplication and is an algebra with identity $p .$ The Hecke algebra of the pair $(B, p)$ is the algebra $p B p .$

Let $B$ be an algebra and let $p$ be an idempotent in $B .$ Then ${End}_{B} (B p) = p B p,$ where $p B p$ acts on $B p$ on the right. More precisely, ${End}_{B} (B p) = \{φ_{h} | h \in p B p\}, where φ_{h} is given by φ_{h} (b p) = b p h, for all b p \in B p .$

		Proof.
	Let $φ \in {End}_{B} (B p)$ and let $b p \in B p$ be such that $φ (p) = b p .$ For all $b' p \in B p, φ (b' p) = φ (b' p p) = b' p φ (p) = b' p b p = (b' p) (p b p),$ and so $φ = φ_{h}$ with $h = p b p .$ The multiplication of the $φ_{h} \in {End}_{B} (B p)$ corresponds to the action of $p B p$ on $B p$ on the right since $(φ_{h_{1}} \circ φ_{h_{2}}) (b p) = ((b p h_{2}) h_{1}) = (b p) (h_{2} h_{1}),$ for all $h_{1}, h_{2} \in p B p$ and $b p \in B p .$ $□$

Recall that if $A$ is a subalgebra of an algebra $B$ and $p$ is an idempotent in $A$ then the left ideal $A p$ is an $A$ -module and

{Ind}_{A}^{B} (A p) ≅ B p .

Haar measure on $G$

Let $G$ be a group. The vector space $ℱ$ of functions $f : G \to ℂ$ is a $G$ -module with $G$ action $(g \cdot f) (h) = f (g^{- 1} h), for all g, h \in G, f \in ℱ .$ Fix a $G$ -submodule $C (G)$ of the space $ℱ$ of all functions on $G .$ A Haar measure on $G$ is a linear functional $μ : C (G) \to ℂ$ such that

(continuity) $μ$ is continuous with respect to the topology on $C (G)$ give by ${∥ f ∥}_{\infty} = \sup \{|f (g)| | g \in G\},$
(positivity) If $f$ is such that $f (g) \in ℝ_{\geq 0}$ for all $g \in G$ then $μ (f) \in ℝ_{\geq 0} .$
(left invariance) For all $g \in G$ and $f \in C (G), μ (g \cdot f) = μ (f) .$

Use the notation

μ (f) = \int_{G} f (g) d μ (g) .

The left invariance of

μ

means that

\int_{G} f (g) d μ (g) = μ (f) = μ (h \cdot f) = \int_{G} f (h^{- 1} g) d μ (g) = \int_{G} f (k) d μ (h k) .

In general it is not true for a Haar measure

μ

that

\int_{G} f (g) d μ (g) = \int_{G} f (g) d μ (g h) or \int_{G} f (g^{- 1}) d μ (g) = \int_{G} f (g) d μ (g),

hold for all

f \in C (G) .

A group

G

is unimodular if one of the euivalent conditions in ??? hold.

The convolution of $f_{1}, f_{2} \in C (G)$ is the function $f_{1} * f_{2}$ on $G$ given by

f_{1} * f_{2} (h) = \int_{G} f_{1} (h g) f_{2} (g^{- 1}) d μ (g), for all h \in G .

Assume that $C (G)$ is closed under concolution and that Fubini's theorem holds.

$C (G)$ is an associative algebra.
If $G$ is unimodular then the map $\vec{t} : C (G) \to ℂ$ given by $\vec{t} (f) = f (1)$ is a trace on $C (G) .$

		Proof.
	Let $f_{1}, f_{2}, f_{3} \in C (G) .$ Then $\begin{array}{rcl} ((f_{1} * f_{2}) * f_{3}) (h) & = & \int_{G} (f_{1} * f_{2}) (h g_{1}) f_{3} ({g_{1}}^{- 1}) d μ (g_{1}) \\ = & \int_{G} \int_{G} f_{1} (h g_{1} g_{2}) f_{2} ({g_{2}}^{- 1}) f_{3} ({g_{1}}^{- 1}) d μ (g_{2}) d μ (g_{1}) \end{array}$ and $\begin{array}{rcl} (f_{1} * (f_{2} * f_{3})) (h) & = & \int_{G} f_{1} (h k) f_{2} * f_{3} (k^{- 1}) d μ (k) \\ = & \int_{G} f_{1} (h k) \int_{G} f_{2} (k^{- 1} g_{1}) f_{3} ({g_{1}}^{- 1}) d μ (g_{1}) d μ (k) \\ = & \int_{G} \int_{G} f_{1} (h g_{1} g_{2}) \int_{G} f_{2} ({g_{2}}^{- 1}) f_{3} ({g_{1}}^{- 1}) d μ (g_{1}) d μ (g_{1} g_{2}) \\ = & \int_{G} \int_{G} f_{1} (h g_{1} g_{2}) f_{2} ({g_{2}}^{- 1}) f_{3} ({g_{1}}^{- 1}) d μ (g_{1}) d μ (g_{2}), \end{array}$ where the last equality is a consequence of the left invariance of $μ .$ Thus, by Fubini's theorem, $(f_{1} * f_{2}) * f_{3} = f_{1} * (f_{2} * f_{3}) .$ Since $\vec{t} (f_{1} * f_{2}) = (f_{1} * f_{2}) (1) = \int_{G} f_{1} (p) f_{2} (p^{- 1}) d μ (p),$ $\vec{t} (f_{2} * f_{1}) = (f_{2} * f_{1}) (1) = \int_{G} f_{2} (p) f_{1} (p^{- 1}) d μ (p),$ and $G$ is unimodular, $\vec{t} (f_{1} * f_{2}) = \vec{t} (f_{2} * f_{1}) . □$

Remark. The algebra $C (G)$ has an identity only when $G$ is finite.

The proof of the following proposition was contributed by Sarah Witherspoon.

Let $H_{1}$ and $H_{2}$ be subgroups of $G$ and let $W_{1}$ and $W_{2}$ be representations of $H_{1}$ and $H_{2}$ respectively. Define $ℋ = \{f : G \to Hom (W_{1}, W_{2}) | f (h_{1} g h_{2}) = W_{1} (h_{1}) f (g) W_{2} (h_{2}), h_{1} \in H_{1}, h_{2} \in H_{2}\} .$ Then ${Hom}_{G} ({Ind}_{H_{1}}^{G} (W_{1}), {Ind}_{H_{2}}^{G} (W_{2})) ≅ ℋ .$

		Proof.
	${Hom}_{G} ({Ind}_{H_{1}}^{G} (W_{1}), {Ind}_{H_{2}}^{G} (W_{2})) ≅ {Hom}_{H_{1}} (W_{2}, {Res}_{H_{1}}^{G} {Ind}_{H_{2}}^{G} (W_{1}))$ Then define $\begin{array}{rcl} {Hom}_{H_{1}} (W_{1}, {Ind}_{H_{2}}^{G} (W_{2})) & \to & ℋ \\ \begin{array}{rcl} ψ : & W_{1} λ & \to & {Ind}_{H_{2}}^{G} (W_{2}) \\ w_{1} & \mapsto & ψ_{w_{1}} \end{array} & \mapsto & T : G \to Hom (W_{1}, W_{2}) \end{array}$ by $ψ_{w_{1}} (g) = T (g) w_{1} .$ Then $(h_{1} ψ_{w_{1}}) (g) = ψ_{w_{1}} (g h_{1}) = T (g h_{1}) w_{1}, and ψ_{h_{1} w_{1}} (g) = T (g) h_{1} w_{1} .$ So $ψ \in {Hom}_{H_{1}} (W_{1}, {Ind}_{H_{2}}^{G} (W_{2}))$ iff $T$ is right invariant undert $H_{1} .$ Also, $ψ_{w_{1}} (h_{2} g) = T (h_{2} g) w_{1} and h_{2} ψ_{w_{1}} (g) = h_{2} T (g) w_{1},$ so $ψ_{w_{1}} \in {Ind}_{H_{2}}^{G} (W_{2})$ iff $T$ is left invariant under $H_{2} . □$

The Hecke algebra $H (G, B, χ)$

Fix a group $G$ and a subgroup $B$ and suppose that $μ : G \to ℂ$ is a Haar measure on $G .$ Let $χ : B \to ℂ^{*}$ be a character of $B .$ Define $C (G / B) = \{f \in C (G) | f (g b) = f (g) χ (b) for all b \in B, g \in G\},$ $C (B \ G / B) = \{f \in C (G) | f (b_{1} g b_{2}) = χ (b_{1}) f (g) χ (b_{2}) for all b_{1}, b_{2} \in B, g \in G\} .$ The following proposition puts these objects into the context of Section ???. It shows that $C (B \ G / B)$ is an algebra under convolution and that $C (G / B)$ is a right $C (B \ G / B)$ -module. The Hecke algebra of the pair $(G, B)$ is the algebra $C (B \ G / B) .$

Define a function $p : G \to ℂ$ by $p (g) = \{\begin{array}{rcl} \frac{χ (g)}{μ (B)}, & if g \in B, \\ 0, & otherwise. \end{array}$

$p$ is an idempotent in $C (G),$
$C (G / B) = C (G) p,$
$C (B \ G / B) = p C (G) p .$

		Proof.
	If $h \in G,$ $(p * p) (h) = \int_{G} p (h k) p (k^{- 1}) d μ (k) = \{\begin{array}{rcl} 0 & if h \notin B, \\ (\frac{χ (h)}{{μ (B)}^{2}}) μ (B), & if h \in B, \end{array} = p (h),$ since $μ (B) = 1 .$ Let $f \in C (G)$ and let $h \in G, b \in B .$ Then, by the left invariance of $μ,$ $\begin{array}{rcl} (f * p) (h b) & = & \int_{G} f (h b k) p (k^{- 1}) d μ (k) = \int_{G} f (h g) p (g^{- 1} b) d μ (b^{- 1} g) \\ = & χ (b) \int_{G} f (h g) p (g^{- 1}) d μ (g) = (f * p) (h) χ (b) . \end{array}$ So $f * p \in C (G \ B)$ and thus $C (G) * p \subseteq C (G / B) .$ Let $f \in C (G / B)$ and $h \in G .$ Then $(f * p) (h) = \int_{G} f (h k) p (k^{- 1}) d μ (k) = \int_{G} f (h) χ (k) χ (k^{- 1}) d μ (k) = f (h) μ (B) = f (h) .$ So $f = f * p \in C (G) * p .$ Thus $C (G / B) \subseteq C (G) * p .$ The proof of (c) is similar to that of (b). $□$

Assume that

Each function in $C (G / B)$ is supported by only a finite number of cosets of $B,$
Each function in $C (B \ G / B)$ is supported by only a finite number of cosets $B w B$ in $B \ G / B,$
For each $B w B \in B \ G / B,$ the number of cosets in $B w B$ is finite.

Fix a set

R

of coset representatives of

G / B

and a subset

W \subseteq R

of coset representatives of

B \ G / B .

For each

x \in R

and each

w \in W

define functions

p_{x} \in C (G / B)

and

T_{w} \in C (B \ G / B)

p_{x} (g) = \{\begin{array}{rcl} \frac{χ (b)}{μ (B)}, & if g = x b \in x B, \\ 0, & otherwise, \end{array} and T_{w} (g) = \{\begin{array}{rcl} \frac{χ (b_{1}) χ (b_{2})}{μ (B)}, & if g = b_{1} w b_{2} \in B w B, & 0, & otherwise. \end{array}

Then

C (G / B) has basis \{p_{x} | x \in R\}, and C (B \ G / B) has basis \{T_{w} | w \in W\} .

The following proposition completely determines the structure of the Hecke a;gebra

C (B \ G / B)

and its action on

C (G / B)

in terms of combinatorics of cosets.

If $x \in R$ and $w \in W$ then $p_{x} * T_{w} = \sum_{y B \in G / B, y B \in x B w B} μ (B) T_{w} (x^{- 1} y) p_{y} .$
If $v, w \in W$ then $T_{u} * T_{v} = \sum_{w \in W} c_{u, v}^{w} T_{w}, where c_{u, v}^{w} = μ {(B)}^{2} \sum_{x B \in B u B \cap w B v^{- 1} B} T_{u} (x) T_{v} (x^{- 1} w) .$

		Proof.
	Let $y \in R .$ Then $\begin{array}{rcl} (p_{x} * T_{w}) (y) & = & \int_{G} p_{x} (g) T_{w} (g^{- 1} y) d μ (g) = \int_{B} p_{x} (x b) T_{w} (b^{- 1} x^{- 1} y) d μ (x b) \\ = & \{\begin{array}{rcl} p_{x} (x) T_{w} (x^{- 1} y) μ (B), & if x^{- 1} y \in B w B, \\ 0, & otherwise,, \end{array} \\ = & \{\begin{array}{rcl} μ (B) T_{w} (x^{- 1} y) p_{y} (y), & if y B \in x B w B, \\ 0, & otherwise., \end{array} \end{array}$ Let $w \in W .$ Then $(T_{u} * T_{v}) (w) = \int_{G} T_{u} (x) T_{v} (x^{- 1} w) d μ (x) = μ (B) \sum_{x B \in w B v^{- 1} B \cap B u B} T_{u} (x) T_{v} (x^{- 1} w)$ $= μ {(B)}^{2} (\sum_{x B \in w B v^{- 1} B \cap B u B} T_{u} (x) T_{v} (x^{- 1} w)) T_{w} (w) . □$

By ??? the map $\vec{t} : C (G) \to C (G)$ given by $\vec{t} (h) = h (1), h \in C (G),$ is a trace on $C (G)$ and, by restriction, $\vec{t}$ is a trace on $C (B \ G / B) .$ Let $⟨,⟩$ be a bilinear form on $C (B \ G / B)$ given by $⟨h_{1}, h_{2}⟩ = \vec{t} (h_{1} h_{2}), h_{1}, h_{2} \in C (B \ G / B) .$

Let $W$ be a set of coset representatives of the cosets in $B \ G / B$ and define $ind (w) = μ (B w B) = (number of cosets of B in B w B),$ for each $w \in W .$ Then

${\{\frac{T_{w^{- 1}}}{ind (w)}\}}_{w \in W}$ is the dual basis to ${\{T_{w}\}}_{w \in W}$ with respect to $⟨,⟩ .$
If $|G \ B|$ is finite, then $\vec{t} = \frac{1}{|G / B|} tr,$ where $tr$ is the trace of the action of $C (B \ G / B)$ on $C (G \ B) .$

		Proof.
	If $u, v \in W$ then $\vec{t} (T_{u} T_{v}) = \sum_{w \in W} c_{u, v}^{w} \vec{t} (T_{w}) = c_{u, v}^{1} = \{\begin{array}{rcl} ind (u), & if B u B = B v^{- 1} B, \\ 0, & otherwise. \end{array}$ If $w \in W,$ $\begin{array}{rcl} tr (T_{w}) & = & \sum_{x B \in G / B} (p_{x} * T_{w}) \|_{p_{x}} = \sum_{x B \in G / B} \sum_{y B \in x B w B} T_{w} (x^{- 1} y) p_{y} \|_{p_{x}} \\ = & \{\begin{array}{cl} \sum_{x B \in G / B} 1, & if w = 1, \\ 0, & otherwise, \end{array} = \{\begin{array}{cl} \|G \ B\| & if w = 1, \\ 0, & otherwise. \end{array} \end{array} □$

Examples

Let $G$ be a finite group. Then the delta function $δ_{g}, g \in G,$ given by $δ_{g} (h) = \{\begin{array}{rcl} 1, & if g = h, \\ 0, & otherwise, \end{array} for all h \in G,$ form a basis of the space $C (G)$ of all functions on $G .$ If $μ$ is a Haar measure on $C (G)$ then $μ (f) = μ (\sum_{g \in G} f (g) δ_{g}) = \sum_{g \in G} f (g) μ (g δ_{1}) = μ (δ_{1}) \sum_{g \in G} f (g),$ and so there is, up to multiplication by constants, a unique Haar measure on the space of all functions on $G .$ Choosing $μ (δ_{1}) = 1 / |G|$ normalises $μ$ so that $μ (G) = 1 .$ The vector space $C (G)$ is closed under convolution with respect to $μ$ and the associative algebra $ℱ$ is isomorphic to the group algebra $ℂ G$ of $G$ via the map $\begin{array}{rcl} C (G) & \to & ℂ G \\ f & \mapsto & μ (δ_{1}) \sum_{g \in G} f (g) g . \end{array}$
Let $G = {GL}_{2} {(ℚ)}^{+} = \{(\begin{array}{rcl} a & b \\ c & d \end{array}) | a, b, c, d \in ℚ, a d - b c > 0\},$ $B = {SL}_{2} (ℤ) = \{(\begin{array}{rcl} a & b \\ c & d \end{array}) | a, b, c, d \in ℤ, a d - b c = 1\} .$ Then the matrices in $W = \{(\begin{array}{rcl} d_{1} & 0 \\ 0 & d_{2} \end{array}) | d_{1}, d_{2} \in ℚ, d_{1} / d_{2} \in ℤ_{> 0}\}$ are a set of coset representatives of the cosets in $B \ G / B .$ The Hecke algebra of the pair $(G, B)$ is commutative and acts on the space of modular forms of weight $k$ on the upper half plane.
Let $𝔽_{q}$ be a finite field with $q$ elements and let $G = {GL}_{n} (𝔽_{q}) = \{A \in M_{n} (𝔽_{q}) | \det (A) \neq 0\},$ $B = \{upper triangular matrices in {GL}_{n} (𝔽_{q})\} .$ Then the set $W = S_{n} = \{permutation matrices in {GL}_{n} (𝔽_{q})\}$ is a set of coset representatives of the cosets in $B \ G / B$ and the Hecke algebra of the pair $(G, B)$ is a deformation of the group algebra of the symmetric group. If $s_{i} = (i, i + 1)$ is the transposition switching $i$ and $i + 1$ in the symmetric group and $T_{i} = T_{s_{i}} = χ_{B s_{i} B}$ denotes the corresponding basis element of the Hecke algebra then $T_{i} T_{j} = T_{j} T_{i}, if |i - j| > 1,$ $T_{i} T_{i + 1} T_{i} = T_{i + 1} T_{i} T_{i + 1}, 1 \leq i \leq n - 1,$ $T_{i}^{2} = (q - 1) T_{i} + q, 1 \leq i \leq n .$ This is a special case of the next example.
Let $G$ be a Chevalley group over the finite field $𝔽_{q}$ with $q$ elements and let $B$ be a Borel subgroup of $G .$ Then the cosets in $B \ G / B$ are indexed by the elements of the Weyl group $W$ and the Hecke algebras of the pair $(G, B)$ is a deformation of the group algebra of $W .$ These algebras were introduced by Iwahori in [Iw].
Let $G$ be a $p$ -adic group and let $B$ be an Iwahori subgroup of $G .$ Then the cosets in $B \ G / B$ are indexed by the elements of the affine Weyl group $\tilde{W}$ and the Hecke algebra of the pair $(G, B)$ is a deformation of the group algebra of $\tilde{W} .$ These algebras were introduced by Iwahori and Matsumoto in [IM].

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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