Operator Valued Modular Spherical Functions

We propose a novel framework that unifies two fundamental concepts in mathematics: matrix-valued spherical functions and scalar modular forms. By extending the classical theory of modular forms to the matrix-valued setting, we introduce and study modular spherical functions . These are smooth functions defined on a connected unimodular Lie group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , a compact subgroup <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> and a discrete subgroup <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> , with values in endomorphism spaces of finite-dimensional vector spaces. Modular spherical functions are characterized as eigenfunctions of the algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>K</mml:mi> </mml:msup> </mml:mrow> </mml:math> , consisting of all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> -left invariant and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -right invariant differential operators on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , while satisfying specific transformation properties under the actions of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Γ</mml:mi> </mml:math> . Focusing on the paradigmatic case <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">SL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>ℝ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">SO</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">SL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>ℤ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , we partially extend the characterization of modular spherical functions. We describe these functions in terms of polynomial eigenfunctions in two real variables for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -Laplacian in the upper half-plane. The results include explicit bases for associated function spaces, recurrence relations for orthogonal polynomials, and analytic continuation. This work advances the study of modular spherical functions, opening new avenues in the representation theory of reductive Lie groups, orthogonal polynomials, and modular forms.