The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras

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

Last update: 1 April 2014

List of notations

§ 1.1	$\Delta ,l,W,⟨,⟩,E,X,X\prime \text{;}$ $T_{\lambda }$ $(\lambda \in X),$ $\stackrel{\sim }{W},$ $\stackrel{\sim }{W}\prime \text{;}$ ${\alpha }^v$ (for $\alpha \in \Delta \text{)}$ [NOTE: The reader is urged to identify ${\alpha }^v$ with the notation ${\alpha }^{\vee }$ standardized by Bourbaki [Bou1968]; here ${\alpha }^v$ has been used throughout for printer's convenience, as there is no chance of confusion]; [Transcriber's note: I have used ${\alpha }^{\vee }$ here instead]; ${\alpha }_i,$ ${\lambda }_i,$ $n_i,$ $R_i$ $(1\leqq i\leqq l)\text{;}$ ${\alpha }_0,$
§ 1.2	${\lambda }_0,$ $n_0,$ $J_0\text{;}$ $E^{+}\supset E^{♯}\supset E^{*}\text{;}$ ${\sigma }_0\in W\text{;}$ ${\theta }_j$ $(j\in J_0)\text{;}$ $\Omega =\left\{{\omega }_j\hspace{0.17em}\right|\hspace{0.17em}j\in J_0\},$ ${\Omega }_0=\left\{{\gamma }_j\hspace{0.17em}\right|\hspace{0.17em}j\in J_0\}\text{;}$
§ 1.3	$l\left(\hspace{0.17em}\right)\text{;}$ ${\delta }_{\sigma },$ $I_{\sigma },$ $w_{\sigma }$ $(\sigma \in W)\text{;}$ $W^{♯}\text{;}$ $r=\left|{\Delta }_{+}\right|,$ $h_{\Delta }=\frac{2r}{l}\text{;}$
§ 1.4	${\stackrel{\sim }{W}}^{+}\supset {\stackrel{\sim }{W}}^{♯}\text{;}$ $\delta \in X\text{;}$ $E_{\text{top}}^{*}\text{;}$ $j_0\in J_0,$ $w_0\in {\stackrel{\sim }{W}}_0^{♯},$ ${\omega }_{j_0}\in \Omega \text{;}$
§ 1.5	$E_J,$ ${\stackrel{\sim }{W}}^J,$ ${\stackrel{\sim }{W}}_J\supset {\stackrel{\sim }{W}}_J^{+}\supset {\stackrel{\sim }{W}}_J^{++}\supset {\stackrel{\sim }{W}}_J^{♯}$ $(J⫋\{0,1,2,\dots ,l\})\text{;}$ ${\left(E^{*}\right)}^w$ is called $w\text{-simplex}$ for $w\in \stackrel{\sim }{W}\prime \text{;}$
§ 1.6	partial orders $\stackrel{A}{\leqq }$ and $\stackrel{B}{\leqq }$ on $\stackrel{\sim }{W}\prime ,$ with restriction to ${\stackrel{\sim }{W}}^{+}$ denoted by $\leqq \text{;}$
§ 2.1	$D\left(\hspace{0.17em}\right)\text{;}$ $b_{\sigma }\in \mathbb{Z}$ (for $\sigma \in W\text{)}$ defined by Conjecture I;
§ 4.1	${𝔤}_{ℂ}\text{;}$ $G,B,T\text{;}$ ${𝔤}_K,{𝔟}_K,{𝔥}_K\text{;}$ $U_{\mathbb{Z}}\text{;}$ irreducible $G\text{-module}$ $M_{\lambda }$ $(\lambda \in X^{+})\text{;}$ partial-order $\leqq$ on $X\text{;}$ $\text{char}\hspace{0.17em}M,$
§ 4.2	$M^{\text{Fr}}$ (for arbitrary $G\text{-module}$ $M\text{)}\text{;}$ $X^{♯}\text{;}$
§ 4.4-4.5	"Weyl module" ${\bar{V}}_{\lambda },$ $c_{\lambda \mu },$ ${\gamma }_{\lambda \mu }$ $(\lambda ,\mu \in X^{+})\text{;}$ $c_{\lambda \mu }$ $(\lambda ,\mu \in X^{♯})\text{;}$ partial-order ${\leqq }^{♯}$ on $X^{♯}\text{;}$ $Y_0\text{;}$
§ 6.1-6.4	$\mathbf{u}\text{;}$ $\Lambda \text{;}$ bijection $\pi :X^{♯}\to \Lambda \text{;}$ ${\mathbf{u}}^{-},$ $\mathbf{b}\prime \text{;}$ $Z_{\lambda },$ $Q_{\lambda },$ $d_{\lambda },$ $a_{\lambda }$ (for $\lambda \in \Lambda \text{)}\text{;}$
§ 5.1	$X^{*}\text{;}$ $X_J^{*}$ $\text{(}J⫋\{0,1,2,\dots ,l\}$ here and in the notations below);
§ 5.5	$c_{w\prime w}^J\in \mathbb{Z}$ (for $w\prime ,w\in {\stackrel{\sim }{W}}_J^{++}\text{)}$ defined by Conjecture III in § 5.4;
§ 7.3	$d_w^J\in \mathbb{Z}$ (for $w\in {\stackrel{\sim }{W}}_J^{♯}\text{)}$ defined by Conjecture III' in § 7.1.

Notes and References

This is an excerpt of the paper The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras by Daya-Nand Verma. It appeared in Lie Groups and their Representations, ed. I.M.Gelfand, Halsted, New York, pp. 653–705, (1975).

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