Pro-p-Iwahori Hecke ring and supersingular Open image in new window -representations
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The motivation of this paper is the search for a Langlands correspondence modulo p. We show that the pro-p-Iwahori Hecke ring Open image in new window of a split reductive p-adic group G over a local field F of finite residue field F q with q elements, admits an Iwahori-Matsumoto presentation and a Bernstein Z-basis, and we determine its centre. We prove that the ring Open image in new window is finitely generated as a module over its centre. These results are proved in  only for the Iwahori Hecke ring. Let p be the prime number dividing q and let k be an algebraically closed field of characteristic p. A character from the centre of Open image in new window to k which is “as null as possible” will be called null. The simple Open image in new window -modules with a null central character are called supersingular. When G=GL(n), we show that each simple Open image in new window -module of dimension n containing a character of the affine subring Open image in new window is supersingular, using the minimal expressions of Haines generalized to Open image in new window , and that the number of such modules is equal to the number of irreducible k-representations of the Weil group W F of dimension n (when the action of an uniformizer p F in the Hecke algebra side and of the determinant of a Frobenius Fr F in the Galois side are fixed), i.e. the number N n (q) of unitary irreducible polynomials in F q [X] of degree n. One knows that the converse is true by explicit computations when n=2 , and when n=3 (Rachel Ollivier).
KeywordsPrime Number Local Field Explicit Computation Central Character Irreducible Polynomial
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