Matrix coefficient


In mathematics, a matrix coefficient is a function on a group of a special form, which depends on a linear representation of the group and additional data. For the case of a finite group, matrix coefficients express the action of the elements of the group in the specified representation via the entries of the corresponding matrices.
Matrix coefficients of representations of Lie groups turned out to be intimately related with the theory of special functions, providing a unifying approach to large parts of this theory. Growth properties of matrix coefficients play a key role in the classification of irreducible representations of locally compact groups, in particular, reductive real and p-adic groups. The formalism of matrix coefficients leads to a generalization of the notion of a modular form. In a different direction, mixing properties of certain dynamical systems are controlled by the properties of suitable matrix coefficients.

Definition

A matrix coefficient of a linear representation of a group on a vector space is a function on the group, of the type
where is a vector in, is a continuous linear functional on, and is an element of. This function takes scalar values on. If is a Hilbert space, then by the Riesz representation theorem, all matrix coefficients have the form
for some vectors and in.
For of finite dimension, and and taken from a standard basis, this is actually the function given by the matrix entry in a fixed place.

Applications

Finite groups

Matrix coefficients of irreducible representations of finite groups play a prominent role in representation theory of these groups, as developed by Burnside, Frobenius and Schur. They satisfy Schur orthogonality relations. The character of a representation ρ is a sum of the matrix coefficients fvi,ηi, where form a basis in the representation space of ρ, and form the dual basis.

Finite-dimensional Lie groups and special functions

Matrix coefficients of representations of Lie groups were first considered by Élie Cartan.
Israel Gelfand realized that many classical special functions and orthogonal polynomials are expressible as the matrix coefficients of representation of Lie groups G. This description provides a uniform framework for proving many hitherto disparate properties of special functions, such as addition formulas, certain recurrence relations, orthogonality relations, integral representations, and eigenvalue properties with respect to differential operators. Special functions of mathematical physics, such as the trigonometric functions, the hypergeometric function and its generalizations, Legendre and Jacobi orthogonal polynomials and Bessel functions all arise as matrix coefficients of representations of Lie groups. Theta functions and real analytic Eisenstein series, important in algebraic geometry and number theory, also admit such realizations.

Automorphic forms

A powerful approach to the theory of classical modular forms, initiated by Gelfand, Graev, and Piatetski-Shapiro, views them as matrix coefficients of certain infinite-dimensional unitary representations, automorphic representations of adelic groups. This approach was further developed by Langlands, for general reductive algebraic groups over global fields.