Holomorphic extensions of representations: (I) automorphic functions

By B. Kroetz, R. Stanton. 2002

101 citations recorded by the source corpus.

Let G be a connected, real, semisimple Lie group contained in its complexification GC, and let K be a maximal compact subgroup of G. We construct a KC-G double coset domain in GC, and we show that the action of G on the K-finite vectors of any irreducible unitary representation of G has a holomorphic extension to this domain. For the resultant holomorphic extension of K-finite matrix coefficients we obtain estimates of the singularities at the boundary, as well as majorant/minorant estimates along the boundary. We obtain L ∞ bounds on holomorphically extended automorphic functions on G/K in terms of Sobolev norms, and we use these to estimate the Fourier coefficients of combinations of automorphic functions in a number of cases, e.g. of triple products of Maas forms.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/17ecf7a7d95e4a15cbedd9cfc12e5da9c44f1028

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In Algebra and Number Theory

1. P. Michel
2. Y. Lau
3. Gergely Harcos
4. Jianya Liu
5. Yangbo Ye
6. V. Blomer
7. R. Stanton
8. B. Kroetz

In Discrete Mathematics and Combinatorics

1. Jianya Liu
2. Yangbo Ye
3. Y. Lau
4. P. Michel
5. R. Stanton
6. B. Kroetz
7. Gergely Harcos
8. V. Blomer

In Mathematical Physics

1. P. Michel
2. Y. Lau
3. Gergely Harcos
4. Jianya Liu
5. Yangbo Ye
6. B. Kroetz
7. R. Stanton
8. V. Blomer

Citations in the corpus

• The subconvexity problem for Rankin–Selberg L-functions and equidistribution of Heegner points. II (2006, background): “[S94,BR99, KSt02 ]) and from a(ny) non-trivial approximation to the Ramanujan–Petersson conjecture, by which we mean that the following hypothesis is satisfied for some θ< 1 (in fact, Hθ is now known for θ = 7 64 [KS03]):”
• Holomorphic extensions of representations: (ii) geometry and harmonic analysis (2005, methodology): “We showed in [ KS04 ] that the domain Gexp(i)KC is an open G-KC double coset in”
• A new bound K2/3+ɛ for Rankin-Selberg ℒ-functions for Hecke congruence subgroups (2010, methodology): “6) To control the inner products against V , we invoke Good [7], Theorem 1, for the case of l ≥ 4, and use Krötz and Stanton [21] for other cases.”