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In this article, we investigate Muirhead's classical system of differentialoperators for the hypergeometric function 1F1 of a matrix argument. Weformulate a conjecture for the combinatorial structure of the characteristicvariety of its Weyl closure which is both supported by computational evidenceas well as theoretical considerations. In particular, we determine the singularlocus of this system.

Algebraic Analysis of the Hypergeometric Function 1F1 of a Matrix Argument