Mirković-Vilonen cycles

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

Last updates: 17 February 2011



((t)) ={ a-c t-c + a-c+1 t-c+1 + | ai, -c } | [[t]] ={ a0 + a1t + a1 t2 + | ai }
Let G0(𝔽) denote a symmetrizable Kac-Moody group with generators
x±α(f) and hλ(g) , for αRre+ , λ𝔥, f𝔽, g𝔽×,
and let
G=G0( ((t)) ) and K=G0( [[t]] )
and let
U- be the subgroup of G generated by the x-α(f),
for αRre+ and f ((t)). The coset space
G/K is the loop Grassmannian.

The Cartan and Iwasawa decompositions are

G= λ 𝔥+ K tλ K and G= μ 𝔥 U- tμ K ,
where tμ =hμ (t-1) The MV-cycles of type λ and weight μ are the irreducible components
Zb= Irr ( K tλ K U- tμ K )
The MV-cycles are indexed by MV-polytopes and, by Baumann-Gaussent [BG, Theorem 4.6],
if   b= f i1 c1 f iN cN b+     then   Zb= yi1 (te1 [t-1] c1 × ) yiN (teN [t-1] cN × ) K ,
yi(f) =x-αi (f),        ej= αij, -cj+1 αij+1 -- cN αiN , and
[t-1] c × ={ a-c t-c + + a-2 t-2 + a-1 t-1 | ai, a-c × } .

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

(i1 ,, id | j1,, jd ) such that Zb= { yi1 (a1tj1 ) yid (adtjd ) tμ K | ai }
The character of Zb is
ch(Zb) = (i1 ,, id | j1,, jd ) fi1 fid,
where the sum is over all composition series of Zb. The character ch(Zb) should be an element of the shuffle algebra [N] just as ch(Λb) 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.


[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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