## The Elliptic Weyl character formula: Flag and Schubert varieties

Last update: 02 March 2012

## Flag and Schubert varieties

The basic data is $G a complex reductive algebraic group ∪| B a Borel subgroup ∪| T a maximal torus.$ The $Weyl group is W0=N(T)/T$ and the character lattice and cocharacter lattice are, respectively, $𝔥ℤ* = Hom(T,ℂ×) and 𝔥ℤ = Hom(ℂ×,T),$ where $\mathrm{Hom}\left(H,K\right)$ is the abelian group of algebraic group homomorphisms from $H\to K$ with product given by pointwise multiplication, $\left(\phi \psi \right)\left(h\right)=\phi \left(h\right)\psi \left(h\right).$ The favourite example of this data is $G=GLn(ℂ) ⊇ { * ⋯ * ⋱ ⋮ 0 * } ⊇ { * ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ * }$

A standard parabolic subgroup of $G$ is a subgroup ${P}_{J}\supseteq B$ such that $G/{P}_{J}$ is a projective variety. A parabolic subgroup of $G$ is a conjugate of a standard parabolic subgroup. The These are studied via the Bruhat decomposition $G= ⨆ w∈W0 BwB and G= ⨆ u∈WJ BuPJ$ where An alternative route to the standard parabolic subgroups is to let $J\subseteq \left\{1,2,...,n\right\}$ and let The Schubert varieties are The $T-$fixed points

Let ${P}_{1},...,{P}_{n}$ be the minimal parabolic subgroups Then

(Coxeter) The group ${W}_{0}$ is generated by ${s}_{1},...,{s}_{n}$ with relations where $\pi /{m}_{ij}={𝔥}^{{\alpha }_{i}}\angle {𝔥}^{{\alpha }_{j}}$ is the angle between ${𝔥}^{{\alpha }_{i}}$ and ${𝔥}^{{\alpha }_{j}}$ where

Then is the root subgroup of $G$ corresponding to $-{\alpha }_{i}$ (adjoint action of $T$ on $G$). Then

Let $w\in {W}_{0}$ and let $w={s}_{{i}_{1}}\cdots {s}_{{i}_{l}}$ be a reduced word for $w.$ The Bott tower or Bott-Samelson variety corresponding to ${s}_{{i}_{1}}\cdots {s}_{{i}_{l}}$ is $Pi1×B Pi2×B ⋯Pil/B → Xw↪G/B (xi1(c1)si1,...,xil(cl)silB) ↦ xi1(c1) si1 ⋯ xil(cl) silB$

In summary there are $T-$equivariant maps and

## Notes and References

These notes are taken from notes on the Elliptic Weyl character formula by Nora Ganter and Arun Ram.

References?