Cusp forms

LetG andH ⊂G be two real semisimple groups defined overQ. Assume thatH is the group of points fixed by an involution ofG. Letπ ⊂L ²(H\G) be an irreducible representation ofG and letf επ be aK-finite function. Let Γ be an arithmetic subgroup ofG. The Poincaré seriesP _f(g)=Σ_H∩ΓΓ f(γ{}itg) is an automorphic form on Γ\G. We show thatP _f is cuspidal in some cases, whenH ∩Γ\H is compact. Partially supported by NSF Grant # DMS 9103608.     

Mixed Hodge polynomials of character varieties

We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of : an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n=2.     

Cusp forms and finite subgroups inSL(5, ℂ)

Samara State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 2, pp. 71–73, April–June, 1995.     

Cusp forms for congruence subgroups of and theta functions

In this paper we use theta functions with rational characteristic to construct cusp forms for congruence subgroups of .The action of the quotient group on these forms is conjugate to the linear action of on . We show that these forms are higher-dimensional analogues of the Fricke functions.

     

Toric varieties and modular forms

Let N⊂ℝ^r be a lattice, and let deg:N→ℂ be a piecewise-linear function that is linear on the cones of a complete rational polyhedral fan. Under certain conditions on deg, the data (N,deg) determines a function f:ℌ→ℂ that is a holomorphic modular form of weight r for the congruence subgroup Γ₁(l). Moreover, by considering all possible pairs (N ,deg), we obtain a natural subring ? (l) of modular forms with respect to Γ₁ (l). We construct an explicit set of generators for ? (l), and show that ? (l) is stable under the action of the Hecke operators. Finally, we relate ? (l) to the Hirzebruch elliptic genera that are modular with respect to Γ₁ (l). Oblatum 22-IX-1999 & 18-X-2000?Published online: 5 March 2001     

Topology of character varieties and representations of quivers

In Hausel et al. (2008) [10] we presented a conjecture generalizing the Cauchy formula for Macdonald polynomial. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a punctured Riemann surface of genus g. We proved several results which support this conjecture. Here we announce new results which are consequences of those in Hausel et al. (2008) [10].     

Representation and character varieties of the Baumslag-Solitar groups

Representation and character varieties of the Baumslag–Solitar groups BS(p, q) are analyzed. Irreducible components of these varieties are found, and their dimension is calculated. It is proved that all irreducible components of the representation variety R_n(BS(p, q)) are rational varieties of dimension n², and each irreducible component of the character variety X_n(BS(p, q)) is a rational variety of dimension k ≤ n. The smoothness of irreducible components of the variety R_n^s (BS(p, q)) of irreducible representations is established, and it is proved that all irreducible components of the variety R_n^s (BS(p, q)) are isomorphic to A¹ {0}.     

On Waring's problem for many forms and Grassmann defective varieties

This paper is devoted to the classical Waring type problem for several algebraic forms. Its geometric translation in terms of Grassmann defectivity for projective varieties yields the best possible solution whenever the number of polynomials is greater than the number of variables. In particular, together with a classical theorem of Alessandro Terracini, this result gives a complete answer to Waring's problem for several ternary forms.     

On character sums of binary quadratic forms

We establish character sum bounds of the form

     

Twisted holomorphic forms on generalized flag varieties

In this paper we prove some vanishing theorems for the twisted Dolbeault cohomology of the complete flag varieties associated to a simple, simply connected algebraic group.     

Logarithmic differential forms on varieties with singularities

In the article we introduce the notion of logarithmic differential forms with poles along a Cartier divisor given on a variety with singularities, discuss some properties of such forms, and describe highly efficient methods for computing the Poincaré series and generators of modules of logarithmic differential forms in various situations. We also examine several concrete examples by applying these methods to the study of divisors on varieties with singularities of many types, including quasi-homogeneous complete intersections, normal, determinantal, and rigid varieties, and so on.