Local functional equations associated with decomposable graphs

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  • Representation Theory of Algebraic Groups and Related Topics : Proceedings of the workshop on Representation Theory, September 15,16, 2012 JOSAI UNIVERSITY

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Let G be a graph with n vertexes {v1, . . . , vn} (without multiple edges) and consider the vector space SymG(R) = {X ∈ Symn(R)|Xij = 0 (vi ?? vj)}. Denote by Sym G(R) its dual vector space. With a statistical motivation, Letac and Massam (Ann. of Statistics, 2007) calculated explicitly the Gamma integral attached to the cones of positive definite “matrices” in Sym G(R) and the dual cone in SymG(R) under the condition that G is decomposable. From their result we can derive rather easily the functional equation for the local zeta functions attached to the cones. In this note, we report that the local zeta functions attached to not necessarily definite connected components also satisfy functional equations. The cones for decomposable G are in general not homogeneous and our functional equations can not be obtained from the theory of prehomogeneous vector spaces. Proofs will appear elsewhere.

Representation Theory of Algebraic Groups and Related Topics : Proceedings of the workshop on Representation Theory, September 15,16, 2012 JOSAI UNIVERSITY / edited by Masatoshi IIDA, Takeyoshi KOGISO, Haruko NISHI, Kiyoko NISHIZAWA. The author is partially supported by the grant in aid of scientific research of JSPS No.24540029.

identifier:JOS-13447777-06_59

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