We determine the unitary dual of the geometric graded Hecke algebras with unequal parameters which appear in Lusztig’s classification of unipotent representations for exceptional p-adic groups. The largest such algebra is of type F4. Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent represent...

Serre in the 1970s was the first to formalize such a question on the way to constructing p-adic L-functions, by way of developing the notion of a p-adic modular form to be the p-adic limit of some compatible family of q-expansions of classical modular forms. Katz came along fairly soon afterwards and generalized the theory to a much more geometric context, and showed that Serre’s p-adic forms e...

It is known that the determination of the Iwahori-spherical unitary dual for p-adic groups can be reduced to the classification of unitary representations with real infinitesimal character for the associated Hecke algebras. In this setting, I determine the Iwahori–spherical unitary dual for split

This paper connects the vanishing at the central critical value of the Lfunctions of certain polarized regular motives with the positivity of the rank of the associated p-adic (Bloch-Kato) Selmer groups. For the motives studied it is shown that vanishing of the L-value implies positivity of the rank of the Selmer group. It is further shown that if the the order of vanishing is positive and even...

The determination of the Iwahori-spherical unitary representations for split p-adic groups can be reduced to the classification of unitary representations with real infinitesimal character for associated graded Hecke algebras. We determine the unitary modules with real infinitesimal character for the graded Hecke algebra of type E6.

The determination of the Iwahori-spherical unitary representations for split p-adic groups can be reduced to the classification of unitary representations with real infinitesimal character for associated graded Hecke algebras. We determine the unitary modules with real infinitesimal character for the graded Hecke algebra of type E6.

1. Iwasawa functions, p-adic measures, and distributions 1 1.1. Some preliminaries 1 1.2. Iwasawa functions and the Iwasawa algebra 3 1.3. Iwasawa functions and interpolation 4 1.4. p-adic distributions and p-adic measures 8 2. The p-adic L-functions of Dirichlet characters 10 2.1. Preliminaries 10 2.2. Iwasawa’s construction 12 3. The p-adic L-functions of modular forms 15 3.1. Preliminaries 1...