Poincaré series on bounded symmetric domains

Abstract

Abstract. We show that any holomorphic automorphic form of sufficiently large weight on an irreducible bounded symmetric domain in C n , n > 1, is the Poincaré series of a polynomial in z 1 ,. . . ,z n and give an upper bound for the degree of this polynomial. We also give an explicit construction of a basis in the space of holomorphic automorphic forms.

On vector-valued automorphic forms on bounded symmetric domains

“…Let us now generalize the construction from [14] by associating to a point p ∈ D m vector-valued Poincaré series (7) Θ (j;k)…”

“…Poincaré series is a standard and powerful tool that is used in automorphic forms, spectral theory, complex analysis, Teichmüller theory, algebraic geometry, and other areas. In [14,15] T. Barron (Foth) studied automorphic forms for compact smooth M = Γ\D (in [15] D = B n ), constructing explicitly the automorphic form f p (p ∈ D) with the property (g, f p ) = g(p) for any other holomorphic automorphic form g. Here (., .) denotes the Petersson inner product.…”

“…There are somewhat different, but closely related results in literature: it is known that Cvalued Poincaré series of polynomials in z 1 ,...,z n span the space of holomorphic automorphic forms on a bounded symmetric domain (for sufficiently large weights) [6,14,42]. In David Bell's thesis [6] it is stated that similar results also hold for vector-valued automorphic forms on classical domains and it is explained how the proofs for C-valued case can be extended to the vector-valued case.…”