Lusztig-Nakajima variety notes from March 2002

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

Last update: 3 September 2013

Lusztig-Nakajima variety notes from March 2002

Let v=iIviki and λ=iIλiωi be the elements of Q+=iI 0αi andP+= iI 0ωi, respectively. Fix I-graded vector spaces V and W with dim(Vi)=vi anddim(Wi)= λi.

Define Ev,w = τΩ± Hom(Vout(τ),Vin(τ)) ( iI Hom(Vi,Wi) Hom(Wi,Vi) ) 𝔤𝔩V = iI𝔤𝔩(Vi) and define the moment map μ:EV,W𝔤𝔩V by μ(x+φ+ψ)i= τΩ+out(τ)=i xτxτ- τΩ+in(τ)=i xτxτ+ iIψiφi.

A point (x+φ+ψ)μ-1(0) is stable if {x-stableSkerφ} ={0}. Pictorially, xW1W1 xW2W1 xW3W1 xW4W1 xW5W1 ψ1φ1 ψ2φ2 ψ3φ3 ψ4φ4 ψ5φ5 0V1V1 0V2 0V3V1 0V4 0V5 0V6V1 ψ6φ6 xW6W1

Let GLV=iIGL(Vi) and define 𝔪(λ)= vQ+𝔪 (v,λ),where 𝔪(v,λ)= μ-1(0)sGLV is the set of GLV-orbits of stable points in μ-1(0).

Define 𝔪0(v,λ)= μ-1(0)sGLV to be the affine variety with coordinate ring given by the space of GLV-invariant polynomials on μ-1(0). Use the map π: 𝔪(v,λ) 𝔪0(v,λ) [x+φ+ψ] the unique closed orbit in GLV·(x+φ+ψ) to define (v,λ)= π-1(0) and (λ)= vQ+ (v,λ). Let λ(1),λ(2)P+ be such that λ(1)+ λ(2)=λ, and fix a decomposition of W=W(1)W(2) so that dim(Wi(1))=λi(1) and dim(Wi(2))=λi(2). Let GLW(1)= iIGL (Wi(1)) and GL(W(2))= iI GL(Wi(2)) and define a one parameter subgroup of GL(W) by λ: * GL(W) t idW(1) tidW(2). Define 𝔷(λ(1),λ(2)) = { [x+φ+ψ]𝔪 (λ)| (limt0λ(t)) (x+φ+ψ) (λ(1))× (λ(2)) } = { [x+φ+ψ]𝔪(λ) |limt0 ( λ(t)π (x+φ+ψ) ) =0 } . Let U be the I-graded vector space given by Ui=Wi ( (ji)Ω± Vj ) and form VσU τV, where σi= (in(τ)=ixτ) +φiand τi=in(τ)=τ xτ-out(τ)=i xτ+ψi. Let bIrr(𝔷) and define εi(b) = dim(Viimτi) φi(b) = dim(kerτiimσi) wt(b) = λ-v,forb Irr(𝔷𝔪(v,λ)) where the sequence (*) is taken with respect to a generic point [x+φ+ψ]b.

Let bx+φ+ψ denote the irreducible component of a point [x+φ+ψ]𝔪(v,λ). Define 𝔪i,(v,λ)= { [x+φ+λ] 𝔪(v,λ)| εi(bx+φ+ψ) = } and consider the maps 𝔪i,εi(b) (v,λ) p 𝔪i,0 (v-εi(b)αi,λ) p 𝔪i,εi(b)-1 (v-αi,λ) p p 𝔪i,εi(b)+1 (v+αi,λ) where p exists if εi(b)>0 and p exists if φi(b)>0.

Then define ei(b)= { p+-1 ( p ( b𝔪i,εi(b) (v,λ) ) ) , ifεi(b)>0, 0, otherwise, and fi(b)= { p--1 ( p ( b𝔪i,εi(b) (v,λ) ) ) , ifφi(b)>0, 0, otherwise.

(a) Irr(𝔷) with wt,εi,φi,ei,fi is a crystal which is isomorphic to B(λ(1))B(λ(2)).
(b) The subset Irr((λ))Irr(𝔷) is a subcrystal isomorphic to B(λ).


[Nak2002] H. Nakajima, Quiver varieties and tensor products, arXiv:math/0103008v2 [math.QA]

[Nak1998] H. Nakajima, Quiver varieties and Kac-Moody algebras, Duke Math, J. 91 No. 3 (1998), 515-560.

[Lus1990-3] G. Lusztig, Canonical bases arising from quantized enveloping algebras II, Common Trends in Mathematics and quantum field theories (T. Eguchi et. al. eds) Progr. Theoret. Phys. Suppl., 102 (1990), pp. 175–201

[KSa1997-2] M. Kashiwara and Y. Saito, Geometric constructions of crystal bases, Duke Math. 89 (1997), 9-36. arXiv:q-alg/9606009

[Sai2000] Y. Saito, Geometric construction of crystal bases II, preprint 2000, arXiv:math/0111232v1 [math.QA]

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