Mirković-Vilonen cycles

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

Last updates: 17 February 2011

MV-cycles

Let

$\begin{array}{rl}ℂ\left(\right(t\left)\right)& =\{a_{-c}{t}^{-c}+a_{-c+1}{t}^{-c+1}+\cdots |a_i\in ℂ,-c\in \mathbb{Z}\}\\ \cup |& \\ ℂ\left[\right[t\left]\right]& =\{a_0+a_{1}t+a_1{t}^2+\cdots |a_i\in ℂ\}\end{array}$

Let $G_0\left(𝔽\right)$ denote a symmetrizable Kac-Moody group with generators

$x_{\pm \alpha }\left(f\right)\text{and}{h}_{{\lambda }^{\vee }}\left(g\right),\text{for}\alpha \in R_{\mathrm{re}}^{+},{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}},f\in 𝔽,g\in {𝔽}^{\times },$

and let

$G=G_0\left(ℂ\right(\left(t\right)\left)\right)\text{and}$ K=G_0\left(ℂ\right[\left[t\right]\left]\right)$$

and let

$U^{-}$ be the subgroup of $G$ generated by the $x_{-\alpha }\left(f\right),$

for $\alpha \in R_{\mathrm{re}}^{+}$ and $f\in ℂ\left(\right(t\left)\right)$. The coset space

$G/K$ is the loop Grassmannian.

The Cartan and Iwasawa decompositions are

$G=\underset{{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}^{+}}{⨆}K{t}_{{\lambda }^{\vee }}K\text{and}G=\underset{{\mu }^{\vee }\in {𝔥}_{\mathbb{Z}}}{⨆}{U}^{-}{t}_{{\mu }^{\vee }}K$,

where $t_{{\mu }^{\vee }}=h_{{\mu }^{\vee }}\left(t^{-1}\right)$ The MV-cycles of type ${\lambda }^{\vee }$ and weight ${\mu }^{\vee }$ are the irreducible components

$Z_b=\mathrm{Irr}\left(\overline{K{t}_{{\lambda }^{\vee }}K\cap U^{-}{t}_{{\mu }^{\vee }}K}\right)$

The MV-cycles are indexed by MV-polytopes and, by Baumann-Gaussent [BG, Theorem 4.6],

if   $b={\stackrel{\sim }{f}}_{i_1}^{c_1}\cdots {\stackrel{\sim }{f}}_{i_N}^{c_N}{b}_{+}$     then   $Z_b=\overline{y_{i_1}\left(t^{e_1}{ℂ\left[t^{-1}\right]}_{c_1}^{\times }\right)\cdots y_{i_N}\left(t^{e_N}{ℂ\left[t^{-1}\right]}_{c_N}^{\times }\right)K}$,

where

$y_i\left(f\right)=x_{-{\alpha }_i}\left(f\right)$,        $e_j=⟨{\alpha }_{i_j},-c_{j+1}{\alpha }_{i_{j+1}}-\cdots -c_N{\alpha }_{i_N}⟩,\text{and}$

${ℂ\left[t^{-1}\right]}_c^{\times }=\{a_{-c}{t}^{-c}+\cdots +a_{-2}{t}^{-2}+a_{-1}{t}^{-1}|a_i\in ℂ,a_{-c}\in {ℂ}^{\times }\}$.

Let $Z_b$ be an MV-cycle of dimension $d$. A composition series for $Z_b$ is

$(i_1,\dots ,i_d|j_1,\dots ,j_d)\text{such that}$ Z_b=\stackrel{‾}{\left\{y_{i_1}\right(a_1{t}^{j_1})\cdots y_{i_d}(a_d{t}^{j_d}\left)t_{{\mu }^{\vee }}K\right|a_i\in ℂ\}}$$

The character of $Z_b$ is

$\mathrm{ch}\left(Z_b\right)={\displaystyle \sum _{(i_1,\dots ,i_d|j_1,\dots ,j_d)}{f}_{i_1}\cdots f_{i_d},}$

where the sum is over all composition series of $Z_b$. The character $\mathrm{ch}\left(Z_b\right)$ should be an element of the shuffle algebra $ℂ\left[N\right]$ just as $\mathrm{ch}\left({\Lambda }_b\right)$ is.

Notes and References

This new notion of the character of an MV-cycle is part of joint work with A. Ghitza and S. Kannan on the relationship between MV-cycles and the Borel-Weil-Bott theorem.

References

[GLS] P. Baumann and S. Gaussent, On Mirković-Vilonen cycles and crystal combinatorics, Representation Theory 12 (2008), 83-130, arXiv:math/0606711, MR2390669.

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