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

Last update: 1 April 2014

## List of notations

 § 1.1 $\Delta ,l,W,⟨,⟩,E,X,X\prime \text{;}$ ${T}_{\lambda }$ $\left(\lambda \in X\right),$ $\stackrel{\sim }{W},$ $\stackrel{\sim }{W}\prime \text{;}$ ${\alpha }^{v}$ (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]; ${\alpha }_{i},$ ${\lambda }_{i},$ ${n}_{i},$ ${R}_{i}$ $\left(1\leqq i\leqq l\right)\text{;}$ ${\alpha }_{0},$ § 1.2 ${\lambda }_{0},$ ${n}_{0},$ ${J}_{0}\text{;}$ ${E}^{+}\supset {E}^{♯}\supset {E}^{*}\text{;}$ ${\sigma }_{0}\in W\text{;}$ ${\theta }_{j}$ $\left(j\in {J}_{0}\right)\text{;}$ $\Omega =\left\{{\omega }_{j} | j\in {J}_{0}\right\},$ ${\Omega }_{0}=\left\{{\gamma }_{j} | j\in {J}_{0}\right\}\text{;}$ § 1.3 $l\left( \right)\text{;}$ ${\delta }_{\sigma },$ ${I}_{\sigma },$ ${w}_{\sigma }$ $\left(\sigma \in W\right)\text{;}$ ${W}^{♯}\text{;}$ $r=|{\Delta }_{+}|,$ ${h}_{\Delta }=\frac{2r}{l}\text{;}$ § 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 ${E}_{J},$ ${\stackrel{\sim }{W}}^{J},$ ${\stackrel{\sim }{W}}_{J}\supset {\stackrel{\sim }{W}}_{J}^{+}\supset {\stackrel{\sim }{W}}_{J}^{++}\supset {\stackrel{\sim }{W}}_{J}^{♯}$ $\left(J⫋\left\{0,1,2,\dots ,l\right\}\right)\text{;}$ ${\left({E}^{*}\right)}^{w}$ 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 $D\left( \right)\text{;}$ ${b}_{\sigma }\in ℤ$ (for $\sigma \in W\text{)}$ defined by Conjecture I; § 4.1 ${𝔤}_{ℂ}\text{;}$ $G,B,T\text{;}$ ${𝔤}_{K},{𝔟}_{K},{𝔥}_{K}\text{;}$ ${U}_{ℤ}\text{;}$ irreducible $G\text{-module}$ ${M}_{\lambda }$ $\left(\lambda \in {X}^{+}\right)\text{;}$ partial-order $\leqq$ on $X\text{;}$ $\text{char} M,$ § 4.2 ${M}^{\text{Fr}}$ (for arbitrary $G\text{-module}$ $M\text{)}\text{;}$ ${X}^{♯}\text{;}$ § 4.4-4.5 "Weyl module" ${\stackrel{‾}{V}}_{\lambda },$ ${c}_{\lambda \mu },$ ${\gamma }_{\lambda \mu }$ $\left(\lambda ,\mu \in {X}^{+}\right)\text{;}$ ${c}_{\lambda \mu }$ $\left(\lambda ,\mu \in {X}^{♯}\right)\text{;}$ 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 ${X}^{*}\text{;}$ ${X}_{J}^{*}$ $\text{(}J⫋\left\{0,1,2,\dots ,l\right\}$ here and in the notations below); § 5.5 ${c}_{w\prime w}^{J}\in ℤ$ (for $w\prime ,w\in {\stackrel{\sim }{W}}_{J}^{++}\text{)}$ defined by Conjecture III in § 5.4; § 7.3 ${d}_{w}^{J}\in ℤ$ (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).