## MV-cycles

Let

 $\begin{array}{rl}ℂ\left(\left(t\right)\right)& =\left\{{a}_{-c}{t}^{-c}+{a}_{-c+1}{t}^{-c+1}+\cdots \phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{a}_{i}\in ℂ,-c\in ℤ\right\}\\ \cup |& \\ ℂ\left[\left[t\right]\right]& =\left\{{a}_{0}+{a}_{1}t+{a}_{1}{t}^{2}+\cdots \phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{a}_{i}\in ℂ\right\}\end{array}$
Let ${G}_{0}\left(𝔽\right)$ denote a symmetrizable Kac-Moody group with generators
 ${x}_{±\alpha }\left(f\right)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{h}_{{\lambda }^{\vee }}\left(g\right),\phantom{\rule{2em}{0ex}}\text{for}\phantom{\rule{.5em}{0ex}}\alpha \in {R}_{\mathrm{re}}^{+},\phantom{\rule{0.5em}{0ex}}{\lambda }^{\vee }\in {𝔥}_{ℤ},\phantom{\rule{0.5em}{0ex}}f\in 𝔽,\phantom{\rule{0.5em}{0ex}}g\in {𝔽}^{×},$
and let
 $G={G}_{0}\left(ℂ\left(\left(t\right)\right)\right)\phantom{\rule{.5em}{0ex}}\text{and}\phantom{\rule{.5em}{0ex}}K={G}_{0}\left(ℂ\left[\left[t\right]\right]\right)$
and let
 ${U}^{-}$ be the subgroup of $G$ generated by the ${x}_{-\alpha }\left(f\right),$
for $\alpha \in {R}_{\mathrm{re}}^{+}$ and $f\in ℂ\left(\left(t\right)\right)$. The coset space
 $G/K$ is the loop Grassmannian.

The Cartan and Iwasawa decompositions are

 $G=\underset{{\lambda }^{\vee }\in {𝔥}_{ℤ}^{+}}{⨆}K{t}_{{\lambda }^{\vee }}K\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}G=\underset{{\mu }^{\vee }\in {𝔥}_{ℤ}}{⨆}{U}^{-}{t}_{{\mu }^{\vee }}K$,
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
 ${Z}_{b}=\mathrm{Irr}\left(\phantom{\rule{.5em}{0ex}}\stackrel{‾}{K{t}_{{\lambda }^{\vee }}K\cap {U}^{-}{t}_{{\mu }^{\vee }}K}\phantom{\rule{.5em}{0ex}}\right)$
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   ${Z}_{b}=\stackrel{‾}{{y}_{{i}_{1}}\left({t}^{{e}_{1}}{ℂ\left[{t}^{-1}\right]}_{{c}_{1}}^{×}\right)\cdots {y}_{{i}_{N}}\left({t}^{{e}_{N}}{ℂ\left[{t}^{-1}\right]}_{{c}_{N}}^{×}\right)K}$,
where
 ${y}_{i}\left(f\right)={x}_{-{\alpha }_{i}}\left(f\right)$,        ${e}_{j}=⟨{\alpha }_{{i}_{j}},-{c}_{j+1}{\alpha }_{{i}_{j+1}}-\cdots -{c}_{N}{\alpha }_{{i}_{N}}⟩,\phantom{\rule{2em}{0ex}}\text{and}$
 ${ℂ\left[{t}^{-1}\right]}_{c}^{×}=\left\{{a}_{-c}{t}^{-c}+\cdots +{a}_{-2}{t}^{-2}+{a}_{-1}{t}^{-1}\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{a}_{i}\in ℂ,{a}_{-c}\in {ℂ}^{×}\right\}$.

Let ${Z}_{b}$ be an MV-cycle of dimension $d$. A composition series for ${Z}_{b}$ is

 $\left({i}_{1},\dots ,{i}_{d}\phantom{\rule{.2em}{0ex}}|\phantom{\rule{.2em}{0ex}}{j}_{1},\dots ,{j}_{d}\right)\phantom{\rule{2em}{0ex}}\text{such that}\phantom{\rule{2em}{0ex}}{Z}_{b}=\stackrel{‾}{\left\{{y}_{{i}_{1}}\left({a}_{1}{t}^{{j}_{1}}\right)\cdots {y}_{{i}_{d}}\left({a}_{d}{t}^{{j}_{d}}\right){t}_{{\mu }^{\vee }}K\phantom{\rule{.5em}{0ex}}|\phantom{\rule{.5em}{0ex}}{a}_{i}\in ℂ\right\}}$
The character of ${Z}_{b}$ is
 $\mathrm{ch}\left({Z}_{b}\right)=\sum _{\left({i}_{1},\dots ,{i}_{d}\phantom{\rule{.2em}{0ex}}|\phantom{\rule{.2em}{0ex}}{j}_{1},\dots ,{j}_{d}\right)}{f}_{{i}_{1}}\cdots {f}_{{i}_{d}},$
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.