Lusztig-Nakajima variety notes from March 2002

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

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 $$Q^{+}=\sum _{i\in I}{\mathbb{Z}}_{\ge 0}{\alpha }_i\text{and}{P}^{+}=\sum _{i\in I}{\mathbb{Z}}_{\ge 0}{\omega }_i,$$ respectively. Fix $I\text{-graded}$ vector spaces $V$ and $W$ with $$\text{dim}\left(V_i\right)=v_i\text{and}\text{dim}\left(W_i\right)={\lambda }_i\text{.}$$

Define $$\begin{array}{ccc}{E}_{v,w}& =& \underset{\tau \in {\Omega }^{\pm }}{⨁}\text{Hom}(V_{\text{out}\left(\tau \right)},V_{\text{in}\left(\tau \right)})\oplus (\underset{i\in I}{⨁}\text{Hom}(V_i,W_i)\oplus \text{Hom}(W_i,V_i))\\ {𝔤𝔩}_V& =& \underset{i\in I}{⨁}𝔤𝔩\left(V_i\right)\end{array}$$ and define the moment map $\mu :E_{V,W}\to {𝔤𝔩}_V$ by $$\mu {(x+\phi +\psi )}_i=\sum _{\underset{\text{out}\left(\tau \right)=i}{\tau \in {\Omega }^{+}}}{x}_{\bar{\tau }}{x}_{\tau }-\sum _{\underset{\text{in}\left(\tau \right)=i}{\tau \in {\Omega }^{+}}}{x}_{\tau }{x}_{\bar{\tau }}+\sum _{i\in I}{\psi }_i{\phi }_i\text{.}$$

A point $(x+\phi +\psi )\in {\mu }^{-1}\left(0\right)$ is stable if $$\{x\text{-stable}\hspace{0.17em}S\subseteq \text{ker}\hspace{0.17em}\phi \}=\left\{0\right\}\text{.}$$ Pictorially, $$\begin{array}{ccccccccccccccccccc}& {}^{\phantom{W_1}}x^{W_1}& & & & {}^{\phantom{W_1}}x^{W_2}& & & & {}^{\phantom{W_1}}x^{W_3}& & & & {}^{\phantom{W_1}}x^{W_4}& & & & {}^{\phantom{W_1}}x^{W_5}& \\ \multicolumn{3}{c}{{\psi }_1⇵{\phi }_1}& & \multicolumn{3}{c}{{\psi }_2⇵{\phi }_2}& & \multicolumn{3}{c}{{\psi }_3⇵{\phi }_3}& & \multicolumn{3}{c}{{\psi }_4⇵{\phi }_4}& & \multicolumn{3}{c}{{\psi }_5⇵{\phi }_5}\\ & {}_{V_1}0_{\phantom{V_1}}& \multicolumn{3}{c}{\leftrightarrows }& \underset{V_2}{0}& \multicolumn{3}{c}{\leftrightarrows }& {}_{\phantom{V_1}}0_{V_3}& \multicolumn{3}{c}{\leftrightarrows }& \underset{V_4}{0}& \multicolumn{3}{c}{\leftrightarrows }& \underset{V_5}{0}\\ & & & & & & & & & ⇵\\ & & & & & & & & & {}_{\phantom{V_1}}0_{V_6}\\ & & & & & & & & \multicolumn{3}{c}{{\psi }_6⇵{\phi }_6}\\ & & & & & & & & & {}_{\phantom{W_1}}x_{W_6}\end{array}$$

Let ${GL}_V=\prod _{i\in I}GL\left(V_i\right)$ and define $$𝔪\left(\lambda \right)=\underset{v\in Q^{+}}{⨆}𝔪(v,\lambda ),\text{where}\hspace{0.17em}𝔪(v,\lambda )=\raisebox{1ex}{${\mu }^{-1}{\left(0\right)}^s$}\!\left/ \!\raisebox{-1ex}{${GL}_V$}\right.$$ is the set of ${GL}_V\text{-orbits}$ of stable points in ${\mu }^{-1}\left(0\right)\text{.}$

Define $${𝔪}_0(v,\lambda )=\raisebox{1ex}{${\mu }^{-1}{\left(0\right)}^s$}\!\left/ \!\raisebox{-1ex}{${GL}_V$}\right.$$ 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 $$\begin{array}{cccc}\pi :& 𝔪(v,\lambda )& ⟶& {𝔪}_0(v,\lambda )\\ & [x+\phi +\psi ]& ⟼& \text{the unique closed orbit in}\hspace{0.17em}\overline{{GL}_V·(x+\phi +\psi )}\end{array}$$ to define $$ℒ(v,\lambda )={\pi }^{-1}\left(0\right)\text{and}ℒ\left(\lambda \right)=\underset{v\in Q^{+}}{⨆}ℒ(v,\lambda )\text{.}$$ Let ${\lambda }^{\left(1\right)},{\lambda }^{\left(2\right)}\in P^{+}$ be such that $${\lambda }^{\left(1\right)}+{\lambda }^{\left(2\right)}=\lambda ,$$ 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 $${GL}_{W^{\left(1\right)}}=\prod _{i\in I}GL\left(W_i^{\left(1\right)}\right)\text{and}GL\left(W^{\left(2\right)}\right)=\prod _{i\in I}GL\left(W_i^{\left(2\right)}\right)$$ and define a one parameter subgroup of $GL\left(W\right)$ by $$\begin{array}{cccc}\lambda :& {ℂ}^{*}& ⟶& GL\left(W\right)\\ & t& ⟼& {\text{id}}_{W^{\left(1\right)}}\oplus t{\text{id}}_{W^{\left(2\right)}}\text{.}\end{array}$$ Define $$\begin{array}{ccc}\stackrel{\sim }{𝔷}({\lambda }^{\left(1\right)},{\lambda }^{\left(2\right)})& =& \{[x+\phi +\psi ]\in 𝔪\left(\lambda \right)\hspace{0.17em}|\hspace{0.17em}\left(\underset{t\to 0}{\text{lim}}\lambda \left(t\right)\right)(x+\phi +\psi )\in ℒ\left({\lambda }^{\left(1\right)}\right)\times ℒ\left({\lambda }^{\left(2\right)}\right)\}\\ & =& \{[x+\phi +\psi ]\in 𝔪\left(\lambda \right)\hspace{0.17em}|\hspace{0.17em}\underset{t\to 0}{\text{lim}}\left(\lambda \left(t\right)\pi (x+\phi +\psi )\right)=0\}\text{.}\end{array}$$ Let $U$ be the $I\text{-graded}$ vector space given by $$U_i=W_i\oplus \left(\underset{\left(ji\right)\in {\Omega }^{\pm }}{⨁}{V}_j\right)$$ and form $$V\stackrel{\sigma }{⟶}U\stackrel{\tau }{⟶}V,$$ where $${\sigma }_i=\left(\underset{\text{in}\left(\tau \right)=i}{⨁}{x}_{\bar{\tau }}\right)+{\phi }_i\text{and}{\tau }_i=\sum _{\text{in}\left(\tau \right)=\tau }{x}_{\tau }-\sum _{\text{out}\left(\tau \right)=i}{x}_{\bar{\tau }}+{\psi }_i\text{.}$$ Let $b\in \text{Irr}\left(\stackrel{\sim }{𝔷}\right)$ and define $$\begin{array}{ccc}{\epsilon }_i\left(b\right)& =& \text{dim}\left(\raisebox{1ex}{$V_i$}\!\left/ \!\raisebox{-1ex}{$\text{im}\hspace{0.17em}{\tau }_i$}\right.\right)\\ {\phi }_i\left(b\right)& =& \text{dim}\left(\raisebox{1ex}{$\text{ker}\hspace{0.17em}{\tau }_i$}\!\left/ \!\raisebox{-1ex}{$\text{im}\hspace{0.17em}{\sigma }_i$}\right.\right)\\ wt\left(b\right)& =& \lambda -v,\text{for}\hspace{0.17em}b\in \text{Irr}(\stackrel{\sim }{𝔷}\cap 𝔪(v,\lambda ))\end{array}$$ where the sequence $(*)$ is taken with respect to a generic point $[x+\phi +\psi ]\in b\text{.}$

Let $b_{x+\phi +\psi }$ denote the irreducible component of a point $[x+\phi +\psi ]\in 𝔪(v,\lambda )\text{.}$ Define $${𝔪}_{i,\ell }(v,\lambda )=\{[x+\phi +\lambda ]\in 𝔪(v,\lambda )\hspace{0.17em}|\hspace{0.17em}{\epsilon }_i\left(b_{x+\phi +\psi }\right)=\ell \}$$ and consider the maps $$\begin{array}{ccccc}{𝔪}_{i,{\epsilon }_i\left(b\right)}(v,\lambda )& \stackrel{p}{⟶}& {𝔪}_{i,0}(v-{\epsilon }_i\left(b\right){\alpha }_i,\lambda )& \stackrel{p\prime }{⟵}& {𝔪}_{i,{\epsilon }_i\left(b\right)-1}(v-{\alpha }_i,\lambda )\\ & & & {}_{p^{\prime \prime }}\nwarrow _{\phantom{p^{\prime \prime }}}\\ & & & & {𝔪}_{i,{\epsilon }_i\left(b\right)+1}(v+{\alpha }_i,\lambda )\end{array}$$ 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 $${\stackrel{\sim }{e}}_i\left(b\right)=\{\begin{array}{cc}\overline{p_{+}^{-1}\left(p(b\cap {𝔪}_{i,{\epsilon }_i\left(b\right)}(v,\lambda ))\right)},& \text{if}\hspace{0.17em}{\epsilon }_i\left(b\right)>0,\\ 0,& \text{otherwise,}\end{array}$$ and $${\stackrel{\sim }{f}}_i\left(b\right)=\{\begin{array}{cc}\overline{p_{-}^{-1}\left(p(b\cap {𝔪}_{i,{\epsilon }_i\left(b\right)}(v,\lambda ))\right)},& \text{if}\hspace{0.17em}{\phi }_i\left(b\right)>0,\\ 0,& \text{otherwise.}\end{array}$$

(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]

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