## Lusztig-Nakajima variety notes from March 2002

Last update: 3 September 2013

## Lusztig-Nakajima variety notes from March 2002

Let $v={\sum }_{i\in I}{v}_{i}{k}_{i}$ and $\lambda ={\sum }_{i\in I}{\lambda }_{i}{\omega }_{i}$ be the elements of $Q+=∑i∈I ℤ≥0αi andP+= ∑i∈I ℤ≥0ωi,$ respectively. Fix $I\text{-graded}$ vector spaces $V$ and $W$ with $dim(Vi)=vi anddim(Wi)= λi.$

Define $Ev,w = ⨁τ∈Ω± Hom(Vout(τ),Vin(τ)) ⊕ ( ⨁i∈I Hom(Vi,Wi)⊕ Hom(Wi,Vi) ) 𝔤𝔩V = ⨁i∈I𝔤𝔩(Vi)$ and define the moment map $\mu :{E}_{V,W}\to {𝔤𝔩}_{V}$ by $μ(x+φ+ψ)i= ∑τ∈Ω+out(τ)=i xτ‾xτ- ∑τ∈Ω+in(τ)=i xτxτ‾+ ∑i∈Iψiφi.$

A point $\left(x+\phi +\psi \right)\in {\mu }^{-1}\left(0\right)$ is stable if ${x-stable S⊆ker φ} ={0}.$ Pictorially, $xW1W1 xW2W1 xW3W1 xW4W1 xW5W1 ψ1⇵φ1 ψ2⇵φ2 ψ3⇵φ3 ψ4⇵φ4 ψ5⇵φ5 0V1V1 ⇆ 0V2 ⇆ 0V3V1 ⇆ 0V4 ⇆ 0V5 ⇵ 0V6V1 ψ6⇵φ6 xW6W1$

Let ${GL}_{V}=\prod _{i\in I}GL\left({V}_{i}\right)$ and define $𝔪(λ)= ⨆v∈Q+𝔪 (v,λ),where 𝔪(v,λ)= μ-1(0)sGLV$ is the set of ${GL}_{V}\text{-orbits}$ of stable points in ${\mu }^{-1}\left(0\right)\text{.}$

Define $𝔪0(v,λ)= μ-1(0)sGLV$ to be the affine variety with coordinate ring given by the space of ${GL}_{V}\text{-invariant}$ polynomials on ${\mu }^{-1}\left(0\right)\text{.}$ Use the map $π: 𝔪(v,λ) ⟶ 𝔪0(v,λ) [x+φ+ψ] ⟼ the unique closed orbit in GLV·(x+φ+ψ)‾$ to define $ℒ(v,λ)= π-1(0) and ℒ(λ)= ⨆v∈Q+ ℒ(v,λ).$ Let ${\lambda }^{\left(1\right)},{\lambda }^{\left(2\right)}\in {P}^{+}$ be such that $λ(1)+ λ(2)=λ,$ and fix a decomposition of $W={W}^{\left(1\right)}\oplus {W}^{\left(2\right)}$ so that $\text{dim}\left({W}_{i}^{\left(1\right)}\right)={\lambda }_{i}^{\left(1\right)}$ and $\text{dim}\left({W}_{i}^{\left(2\right)}\right)={\lambda }_{i}^{\left(2\right)}\text{.}$ Let $GLW(1)= ∏i∈IGL (Wi(1)) and GL(W(2))= ∏i∈I GL(Wi(2))$ and define a one parameter subgroup of $GL\left(W\right)$ by $λ: ℂ* ⟶ GL(W) t ⟼ idW(1)⊕ tidW(2).$ Define $𝔷∼(λ(1),λ(2)) = { [x+φ+ψ]∈𝔪 (λ) | (limt→0λ(t)) (x+φ+ψ)∈ℒ (λ(1))× ℒ(λ(2)) } = { [x+φ+ψ]∈𝔪(λ) | limt→0 ( λ(t)π (x+φ+ψ) ) =0 } .$ Let $U$ be the $I\text{-graded}$ vector space given by $Ui=Wi⊕ ( ⨁(ji)∈Ω± Vj )$ and form $V⟶σU ⟶τV,$ where $σi= (⨁in(τ)=ixτ‾) +φiand τi=∑in(τ)=τ xτ-∑out(τ)=i xτ‾+ψi.$ Let $b\in \text{Irr}\left(\stackrel{\sim }{𝔷}\right)$ and define $εi(b) = dim(Viim τi) φi(b) = dim(ker τiim σi) wt(b) = λ-v,for b∈ Irr(𝔷∼∩𝔪(v,λ))$ where the sequence $\left(*\right)$ is taken with respect to a generic point $\left[x+\phi +\psi \right]\in b\text{.}$

Let ${b}_{x+\phi +\psi }$ denote the irreducible component of a point $\left[x+\phi +\psi \right]\in 𝔪\left(v,\lambda \right)\text{.}$ Define $𝔪i,ℓ(v,λ)= { [x+φ+λ]∈ 𝔪(v,λ) | εi(bx+φ+ψ) =ℓ }$ 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 $p\prime$ exists if ${\epsilon }_{i}\left(b\right)>0$ and ${p}^{\prime \prime }$ exists if ${\phi }_{i}\left(b\right)>0\text{.}$

Then define $e∼i(b)= { p+-1 ( p ( b∩𝔪i,εi(b) (v,λ) ) ) ‾ , if εi(b)>0, 0, otherwise,$ and $f∼i(b)= { p--1 ( p ( b∩𝔪i,εi(b) (v,λ) ) ) ‾ , if φi(b)>0, 0, otherwise.$

 (a) $\text{Irr}\left(\stackrel{\sim }{𝔷}\right)$ with $wt,{\epsilon }_{i},{\phi }_{i},{\stackrel{\sim }{e}}_{i},{\stackrel{\sim }{f}}_{i}$ is a crystal which is isomorphic to $B\left({\lambda }^{\left(1\right)}\right)\otimes B\left({\lambda }^{\left(2\right)}\right)\text{.}$ (b) The subset $\text{Irr}\left(ℒ\left(\lambda \right)\right)\subseteq \text{Irr}\left(\stackrel{\sim }{𝔷}\right)$ is a subcrystal isomorphic to $B\left(\lambda \right)\text{.}$

## References

[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]