Publications

A q-analog of Freudenthal's weight multiplicity formula, Indagationes Mathematicae, N.S., 11(1) (2000), 87-94.
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Abstract: In this paper, we give a formula which calculates certain of Lusztig's q-analogs of weight multiplicity and hence certain Kazhdan-Lusztig polynomials for the affine Weyl group of a semisimple, simply connected algebraic group G over C. It does so recursively with respect to the standard partial ordering on the character module of a maximal torus of G.

Decomposition of double cosets in p-adic groups, Pacific Journal of Mathematics, 197(1) (2001), 97-117.

Abstract:Let G be a split group over a locally compact field F with non-trivial discrete valuation. Employing the structure theory of such groups and the theory of Coxeter groups, we obtain a general formula for the decomposition of double cosets P₁gP₂ of subgroups P₁, P₂ of G(F) containing an Iwahori subgroup into left cosets of P₂. When P₁ and P₂ are the same hyperspecial subgroup, we use this result to derive a formula of Iwahori for the degrees of elements of the spherical Hecke algebra.

Parahoric fixed spaces in unramified principal series representations, Pacific Journal of Mathematics, 204(2) (2002), 433-443.

Abstract: Let k be a non-archimedean locally compact field and let G be the set of k-points of a connected reductive group defined over k. Let W be the relative Weyl group of G, and let H (G,B) be the Hecke algebra of G with respect to an Iwahori subgroup B of G. We compute the effects of H (G,B) and W on the B-fixed vectors of an unramified principal series representation I of G. We use this computation to determine the dimension of the space of K-fixed vectors in I, where K is a parahoric subgroup of G.

Hecke algebras and automorphic forms, with D. Pollack, Compositio Mathematica, 130(1) (2002), 21-48.
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Abstract: The goal of this paper is to carry out some explicit calculations of the action of local Hecke algebras on spaces of algebraic modular forms on certain simple groups. We present the results of our calculations along with interpretations of these results concerning the lifting of forms and Galois representations. The data we have obtained is of interest both from the point of view of number theory and of representation theory. For example, our data, together with a conjecture of Gross, predicts the existence of a Galois extension of Q with Galois group G₂(F₅) which is ramified only at the prime 5. We also provide evidence of the existence of the symmetric cube lifting from PGL₂ to PGSp₄.

On the correspondence of representations between GL(n) and division algebras, with A. Raghuram, Proceedings of the American Mathematical Society, 131 (2003), 1641-1648.

Abstract: For a division algebra D over a p-adic field F, we prove that depth is preserved under the correspondence of discrete series representations of GL_n(F) and irreducible representations of D^* by proving that an explicit relation holds between depth and conductor for all such representations. We also show that this relation holds for many (possibly all) discrete series representations of GL₂(D).

On Conductors and Newforms for U(1,1), with A. Raghuram, Proceedings of the Indian Academy of Science, 114(3) (2004), 1-25.
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Abstract: Let F be a non-Archimedean local field whose residue characteristic is odd. In this paper, building on previous work on SL₂(F), we develop a theory of newforms for U(1,1)(F), the quasi split unramified unitary group in two variables. This is analogous to results of Casselman for GL₂(F) and Jacquet, Piatetski-Shapiro, and Shalika for GL_n(F). A newform of a representation π is a vector in π that is essentially fixed by a certain congruence subgroup. We show that every space of newforms contains test vectors for some standard Whittaker functionals.

Depth-zero base change for unramified U(2,1), with J. Adler, Journal of Number Theory, 114(2) (2005), 324-360. Printer's error corrected in Journal of Number Theory, 121(1) (2006), 186.
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Abstract: We give an explicit description of L-packets and quadratic base change for depth-zero representations of unramified unitary groups in two and three variables. We show that this base change is compatible with unrefined minimal K-types.

A variation on the solvable case of the Dedekind conjecture, with K. Wilson, Journal of the Ramanujan Mathematical Society, 20(2) (2005), 1-10.
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Abstract: Let G be the Galois group of a solvable Galois extension K/F of number fields. In this note, we demonstrate the holomorphy of certain Artin L-functions attached to K/F, generalizing results of M. R. Murty and A. Raghuram. We also give a bound (generalizing one of Murty-Raghuram) on the orders of certain Artin L-functions at an arbitrary point in the complex plane by the order of a corresponding quotient of Dedekind zeta functions. We deduce some corollaries on the possible orders of zeros of such quotients.

Conductors and Newforms for SL(2), with A. Raghuram, Pacific Journal of Mathematics, 231(1) (2007), 127-154.
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Abstract: In this paper we develop a theory of newforms for SL₂(F) where F is a non-Archimedean local field whose residue characteristic is odd. This is analogous to results of Casselman for GL₂(F) and Jacquet, Piatetski-Shapiro, and Shalika for GL_n(F). To a representation π of SL₂(F) we attach an integer c(π), called the conductor of π. The conductor of π depends only on the L-packet Π containing π. It is shown to be equal to the conductor of the minimal representation of GL₂(F) determining Π. A newform is a vector in π that is essentially fixed by a congruence subgroup of level c(π). We show that our newforms are always test vectors for some standard Whittaker functionals, and, in doing so, we give various explicit formulas for newforms.