The Elliptic Weyl character formula: Flag and Schubert varieties

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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, 𝔥* = Hom(T,×) and 𝔥 = Hom(×,T), where Hom(H,K) is the abelian group of algebraic group homomorphisms from HK with product given by pointwise multiplication, (φψ)(h) = φ(h)ψ(h). The favourite example of this data is G=GLn() { * * 0 * } { * 0 0 * }

A standard parabolic subgroup of G is a subgroup PJB such that G/PJ 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/PJ  are the   partial flag varieties. These are studied via the Bruhat decomposition G= wW0 BwB and G= uWJ BuPJ where WJ= {vW0  |  vTPJ} and WJ= {coset representatives  u  of cosets in   W0/WJ}. An alternative route to the standard parabolic subgroups is to let J{1,2,...,n} and let WJ= sj  |  jJ. Then PJ= uWJ BuB. The Schubert varieties are Xw= BwB_   in   G/B and XuJ= BuPL_   in   G/PJ. The T-fixed points is   G/B   are   {wB  |  wW0}, and in   G/PJ   are   {uPJ  |  uWJ}.

Let P1...Pn be the minimal parabolic subgroups ( PiB   and   Pi=P{i} ). Then Wi=W{i}={1,si} and   are the   simple reflections   in   W0.

(Coxeter) The group W0 is generated by with relations si2=1 and sisjsi mij   factors = sjsisj mij   factors where π/mij = 𝔥αi𝔥αj is the angle between 𝔥αi and 𝔥αj where 𝔥αi = {μ𝔥  |  μ,αi=0}.

Then Pi = 𝔛-αi,B, where 𝔛αi = {x-αi(c)  |  c} is the root subgroup of G corresponding to -αi (adjoint action of T on G). Then Pi = BBsiB with BsiB = {xi(c)ni-1B  |  c} where xi(c) = xαi(c).

Let wW0 and let w= si1sil be a reduced word for w. The Bott tower or Bott-Samelson variety corresponding to si1sil is Pi1×B Pi2×B Pil/B XwG/B (xi1(c1)si1,...,xil(cl)silB) xi1(c1) si1 xil(cl) silB

In summary there are T-equivariant maps pJ: G/B G/PJ gB gPJ for   J {1,2,...,l}, (FSv 1) ιw: pt G/B pt wB for   wW0, (FSv 2) σw: Xw G/B gB gB for   wW0, (FSv 3) and ιuJ: pt G/PJ pt uPJ for   uWJ, (FSv 4) σuJ: XuJ G/PJ gPJ gPJ for   uWJ. (FSv 5)

Notes and References

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



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