Maass–Selberg Relations in Harmonic Analysis

• Maass–Selberg relations are fundamental identities in harmonic analysis that define norm and inner product relations among Whittaker integrals and intertwining operators in real reductive groups.
• They emerge from Harish-Chandra’s framework and are established via meromorphic continuation of standard intertwining operators and finite-rank B-matrices.
• These relations ensure the analytic behavior and unitarity of normalized Fourier and Whittaker transforms, impacting automorphic form analysis and wave packet construction.

The Maass–Selberg relations are fundamental identities in the harmonic analysis of real reductive groups, governing the norm and inner product relations among Whittaker integrals and intertwining operators within the generalized principal series. Emerging from Harish-Chandra’s foundational work and thoroughly established for Whittaker functions in van den Ban’s complete proof, these relations provide the analytical backbone for the decent behavior of normalized Whittaker integrals and ensure the regularity properties necessary for rigorous treatments of automorphic Fourier and wave packet transforms (Ban, 24 Nov 2025).

1. Framework and Notation

Let $G$ be a real reductive group of Harish–Chandra class with Iwasawa decomposition $G = K\,A\,N_0$ ($K$ maximal compact, $A \simeq \mathbb{R}^d$, $N_0$ nilpotent). Fix a generic unitary character $\chi : N_0 \to S^1$, and write $\Sigma = \Sigma(\mathfrak{a}, \mathfrak{g})$ for the restricted root system with a chosen set of positive roots $\Sigma^+$ so that $$\operatorname{Lie} N_0 = \bigoplus_{\alpha \in \Sigma^+} \mathfrak{g}_\alpha$$.

Standard parabolic subgroups $\mathcal{P}(A)$ are those containing $A$, each with Langlands decomposition $P = M_P\,A_P\,N_P$, and $2\rho_P \in \mathfrak{a}_P^*$ denoting the sum of positive $P$-roots. For a discrete series representation $(\sigma, H_\sigma)$ of $M_P$ and $\nu \in \mathfrak{a}_P^* \otimes \mathbb{C}$, the normalized induced representation is

$$I(P, \sigma, \nu) = \operatorname{Ind}_{M_P A_P N_P}^G(\sigma \otimes e^\nu \otimes 1),$$

acting on the Fréchet space $C^\infty(G/P:\sigma \otimes \nu)$ of smooth, equivariant functions.

2. Whittaker Distribution Vectors

Distribution vectors in the contragredient $C^{-\infty}(G/P:\sigma,\nu)$ are continuous linear functionals on $C^\infty(G/P:\sigma, -\bar{\nu})$. The subspace of Whittaker type, $C^{-\infty}(G/P:\sigma,\nu)_\chi$, comprises those distributions $\Lambda$ satisfying $\Lambda(ng) = \chi(n)\Lambda(g)$ for $n \in N_0$. When $P$ is opposite-standard, evaluation at the identity establishes an isomorphism

$$ev : C^{-\infty}(G/P:\sigma,\nu)_\chi \to H_\sigma^{-\infty}{}_{\chi_P}$$

where $\chi_P$ is the restriction of $\chi$ to $M_P \cap N_0$ and $j(P, \sigma, \nu)$ denotes its inverse. For general $P$, conjugation allows similar realization via a representative $v_P \in N_K(A)$ such that $v_P P v_P^{-1}$ is opposite. The resulting family $j(P, \sigma, \nu)$ is meromorphic in $\nu$.

3. Standard Intertwining Operators and B-Matrices

For $P_1, P_2 \in \mathcal{P}(A)$ sharing the $A$-component, the standard intertwining operator

$$A(P_2, P_1; \nu): C^\infty(G/P_1:\sigma,\nu) \to C^\infty(G/P_2:\sigma,\nu)$$

is defined for generic $\nu$ by integration over $N_2 \cap \bar{N}_1$, admitting meromorphic continuation. The adjoint relation

$A(P_1, P_2; -\bar{\nu})^* = A(P_2, P_1; \nu)$

holds. The action of $A(P_2, P_1; \nu)$ on Whittaker distribution vectors factors through the finite-rank B-matrices: $$A(P_2,P_1;\nu)\,j(P_1, \sigma, \nu) = j(P_2, \sigma, \nu)\,B(P_2, P_1; \nu)$$ where

$$B(P_2,P_1;\nu) : H^{-\infty}_\sigma{}_{\chi_{P_1}} \to H^{-\infty}_\sigma{}_{\chi_{P_2}}$$

is meromorphic in $\nu$. The normalization factor $\eta(P,Q;\nu)$ is defined by

$$A(P,Q;\nu)\,A(Q,P;\nu) = \eta(P,Q;\nu)\,\operatorname{id},$$

with a corresponding adjoint identity in terms of $A(Q,P;-\bar{\nu})^*$.

4. The Maass–Selberg Relations

For $P, Q \in \mathcal{P}(A)$ with common $A$-component, the Maass–Selberg relation asserts

$$B(Q, P; -\bar{\nu})^*\,B(Q, P; \nu) = \eta(Q, P; \nu)\,\operatorname{id}$$

for all $\nu \in \mathfrak{a}_P^*$. Alternatively,

$B(P, Q; \nu) = B(Q, P; -\bar{\nu})^*.$

These identities constrain the norm and adjoint relations among the B-matrices and play a pivotal role in the normalization and analytic properties of Whittaker integrals.

5. Proof Strategy and Structural Reductions

The proof proceeds by (A) induction on the rank via product decompositions: for successive parabolics $P_1, P_2, P_3$,

$$A(P_3, P_1; \nu) = A(P_3, P_2; \nu)\,A(P_2, P_1; \nu)\quad\Rightarrow\quad B(P_3, P_1;\nu)=B(P_3, P_2;\nu)\,B(P_2, P_1;\nu).$$

Reduction to adjacent parabolic pairs is accomplished by wall-crossing in the Weyl group. Weyl conjugacy, with $w \in W$ sending $P \to P'$, $Q \to Q'$, yields

$B(Q', P'; w\nu) = U_w\,B(Q, P; \nu)\,U_w^{-1}$

where $U_w$ is a unitary twist, reducing the proof to representative configurations per $W$-orbit.

If $P, Q$ are adjacent but not maximal, further reduction proceeds via the centralizer $G' = Z_G(X)$, with $X$ orthogonal to the root separating $P$ from $Q$, and relates B-matrices on $G$ to those on $G'$, of strictly lower rank.

In the basic case (compact center, opposite maximal parabolics), Harish–Chandra’s boundary integral argument applies: considering Whittaker functions subject to Casimir operator action, a boundary bracket integrated over expanding domains $G[t]$ via the radial-part formula and Gauss divergence theorem forces the desired norm relations by vanishing of the total boundary contribution. This confirms the Maass–Selberg relations for normalized Whittaker coefficients, and by backtracking, for all parabolic pairs.

6. Whittaker C–functions and Associated Relations

Given a Whittaker integral $(P, \psi; \nu)$ with $\psi \in L^2(\tau: P)$, the constant term along an associate $Q \sim P$ admits a finite expansion with coefficients $a^{s\nu}\,C_{Q|P}(s;\nu)\psi(m)$, where the operator-valued $C_{Q|P}(s;\nu)$ are Whittaker C-functions. Their relation to the intertwining operators and B-matrices leads to the Maass–Selberg relation at the level of C-functions: $$C_{Q|P}(s; -\bar{\nu})^*\,C_{Q|P}(s; \nu) = \eta(P, \bar{P}; \nu)\,\operatorname{id}_{L^2(\tau:P)}$$ and unitarity in the basic case. These relations are essential in constructing normalized Whittaker integrals and establishing analytic continuations.

7. Functional Consequences for Harmonic Analysis

The normalized Whittaker integral is defined as

$${}^\circ W(P,\psi;\nu)(x) := (P,\,\psi';\nu)(x),\quad \psi' = C_{P|P}(1;\nu)^{-1}\psi,$$

satisfying

$${}_Q^\circ W(P, \psi; \nu) = {}^\circ W(Q, C_{Q|P}^\circ(s;\nu)\psi; s\nu).$$

The Maass–Selberg unitarity for normalized C-matrices implies that for each $P$, $\nu \mapsto {}^\circ W(P, \psi; \nu)$ is holomorphic in a tubular region around $i\mathfrak{a}_P^*$ and exhibits uniform Schwartz-type decay in both $x$ and $\nu$.

The normalized Fourier transform

$${}^\circ\!{\mathcal F}_P : (\tau: G/N_0:\chi) \to \mathcal{S}(i\mathfrak{a}_P^*, L^2(\tau:P)), \qquad ({}^\circ\!{\mathcal F}_P f)(\nu) = \int_{G/N_0}\langle f(x), {}^\circ W(P,\cdot,\nu)(x)\rangle\,dx$$

transfers Schwartz spaces to Euclidean Schwartz spaces in $\nu$. Its adjoint, the wave-packet transform

$${}^\circ\!\mathcal{W}_P: \mathcal{S}(i\mathfrak{a}_P^*, L^2(\tau:P)) \to (\tau: G/N_0:\chi), \qquad ({}^\circ\!\mathcal{W}_P \Phi)(x) = \int_{i\mathfrak{a}_P^*} {}^\circ W(P, \Phi(\nu); \nu)(x)\,d\nu$$

is continuous, reflecting the regularity provided by the Maass–Selberg relations. The functional equations

$${}^\circ W(P,\nu) = {}^\circ W(Q, s\nu)\,C_{Q|P}^\circ(s, \nu),\qquad C_{Q|P}^\circ(s; \nu){}^\circ{\mathcal F}_P f(\nu) = {}^\circ{\mathcal F}_Q f(s\nu)$$

mirror Harish–Chandra's Plancherel equations in the Whittaker setting, confirming the deep structural role of the Maass–Selberg relations in non-spherical harmonic analysis (Ban, 24 Nov 2025).