The Elliptic Weyl character formula: Flag and Schubert varieties

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

Last update: 02 March 2012

Flag and Schubert varieties

The basic data is $\begin{array}{cl} G & a complex reductive algebraic group \\ \cup| \\ B & a Borel subgroup \\ \cup| \\ T & a maximal torus. \end{array}$ The $Weyl group is W_{0} = N (T) / T$ and the character lattice and cocharacter lattice are, respectively, $𝔥_{ℤ}^{*} = Hom (T, ℂ^{\times}) and 𝔥_{ℤ} = Hom (ℂ^{\times}, T),$ where $Hom (H, K)$ is the abelian group of algebraic group homomorphisms from $H \to K$ with product given by pointwise multiplication, $(φ ψ) (h) = φ (h) ψ (h) .$ The favourite example of this data is $G = G L_{n} (ℂ) \supseteq {(\begin{matrix} * & \dots & * \\ ⋱ & ⋮ \\ 0 & * \end{matrix})} \supseteq {(\begin{matrix} * & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & * \end{matrix})}$

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 $flag variety is G / B and G / P_{J} are the partial flag varieties.$ These are studied via the Bruhat decomposition $G = ⨆_{w \in W_{0}} B w B and G = ⨆_{u \in W^{J}} B u P_{J}$ where $W_{J} = {v \in W_{0} | v T \in P_{J}} and W^{J} = {coset representatives u of cosets in W_{0} / W_{J}} .$ An alternative route to the standard parabolic subgroups is to let $J \subseteq {1, 2, ..., n}$ and let $W_{J} = ⟨ s_{j} | j \in J ⟩ . Then P_{J} = ⨆_{u \in W_{J}} B u B .$ The Schubert varieties are $X_{w} = \overline{B w B} in G / B and X_{u}^{J} = \overline{B u P_{L}} in G / P_{J} .$ The $T -$ fixed points $is G / B are {w B | w \in W_{0}}, and in G / P_{J} are {u P_{J} | u \in W^{J}} .$

Let $P_{1} ... P_{n}$ be the minimal parabolic subgroups $(P_{i} \neq B and P_{i} = P_{{i}}) .$ Then $W_{i} = W_{{i}} = {1, s_{i}} and s_{1} ... s_{n} are the simple reflections in W_{0} .$

(Coxeter) The group $W_{0}$ is generated by $s_{1} ... s_{n}$ with relations $s_{i}^{2} = 1 and \underset{m_{i j} factors}{\underset{⏟}{s_{i} s_{j} s_{i} \dots}} = \underset{m_{i j} factors}{\underset{⏟}{s_{j} s_{i} s_{j} \dots}}$ where $π / m_{i j} = 𝔥^{α_{i}} ∠ 𝔥^{α_{j}}$ is the angle between $𝔥^{α_{i}}$ and $𝔥^{α_{j}}$ where $𝔥^{α_{i}} = {μ \in 𝔥_{ℝ} | ⟨ μ, α_{i} ⟩ = 0} .$

Then $P_{i} = ⟨ 𝔛_{- α_{i}}, B ⟩, where 𝔛_{α_{i}} = {x_{- α_{i}} (c) | c \in ℂ}$ is the root subgroup of $G$ corresponding to $- α_{i}$ (adjoint action of $T$ on $G$ ). Then $P_{i} = B \cup B s_{i} B with B s_{i} B = {x_{i} (c) n_{i}^{- 1} B | c \in ℂ} where x_{i} (c) = x_{α_{i}} (c) .$

Let $w \in W_{0}$ and let $w = s_{i_{1}} \dots s_{i_{l}}$ be a reduced word for $w .$ The Bott tower or Bott-Samelson variety corresponding to $s_{i_{1}} \dots s_{i_{l}}$ is $\begin{array}{ccc} P_{i_{1}} \times_{B} P_{i_{2}} \times_{B} \dots P_{i_{l}} / B & \to & X_{w} ↪ G / B \\ (x_{i_{1}} (c_{1}) s_{i_{1}}, ..., x_{i_{l}} (c_{l}) s_{i_{l}} B) & \mapsto & x_{i_{1}} (c_{1}) s_{i_{1}} \dots x_{i_{l}} (c_{l}) s_{i_{l}} B \end{array}$

In summary there are $T -$ equivariant maps $\begin{array}{lrc} \begin{array}{rrcl} p_{J} : & G / B & \to & G / P_{J} \\ g B & \mapsto & g P_{J} \end{array} & for J \subseteq {1, 2, ..., l}, & (FSv 1) \\ \begin{array}{rrcl} ι_{w} : & pt & ↪ & G / B \\ pt & \mapsto & w B \end{array} & for w \in W_{0}, & (FSv 2) \\ \begin{array}{rrcl} σ_{w} : & X_{w} & ↪ & G / B \\ g B & \mapsto & g B \end{array} & for w \in W_{0}, & (FSv 3) \end{array}$ and $\begin{array}{lrc} \begin{array}{rrcl} ι_{u}^{J} : & pt & ↪ & G / P_{J} \\ pt & \mapsto & u P_{J} \end{array} & for u \in W^{J}, & (FSv 4) \\ \begin{array}{rrcl} σ_{u}^{J} : & X_{u}^{J} & ↪ & G / P_{J} \\ g P_{J} & \mapsto & g P_{J} \end{array} & for u \in W^{J} . & (FSv 5) \end{array}$

Notes and References

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

References

References?

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