JournalsjemsVol. 15, No. 4pp. 1375–1455

Quiver varieties and the character ring of general linear groups over finite fields

  • Emmanuel Letellier

    Université de Caen, Caen, France
Quiver varieties and the character ring of general linear groups over finite fields cover
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Abstract

Given a tuple (\calX1,,\calXk)(\calX_1,\dots,\calX_k) of irreducible characters of \GLn(\Fq)\GL_n(\F_q) we define a star-shaped quiver Γ\Gamma together with a dimension vector \v. Assume that (\calX1,,\calXk)(\calX_1,\dots,\calX_k) is \emph{generic}. Our first result is a formula which expresses the multiplicity of the trivial character in the tensor product \calX1\calXk\calX_1\otimes\cdots\otimes\calX_k as the trace of the action of some Weyl group on the intersection cohomology of some (non-affine) quiver varieties associated to (Γ,)ˇ(\Gamma,\v). The existence of such a quiver variety is subject to some condition. Assuming that this condition is satisfied, we prove our second result: The multiplicity \calX1\calXk,1\langle \calX_1\otimes\cdots\otimes\calX_k,1\rangle is non-zero if and only if \v is a root of the Kac-Moody algebra associated with Γ\Gamma. This is somehow similar to the connection between Horn's problem and the representation theory of \GLn(\C)\GL_n(\C)}.

Cite this article

Emmanuel Letellier, Quiver varieties and the character ring of general linear groups over finite fields. J. Eur. Math. Soc. 15 (2013), no. 4, pp. 1375–1455

DOI 10.4171/JEMS/395