Langlands classification

Remarks

Infinitesimal equivalence means that the Harish-Chandra modules are isomorphic. The proof compares the leading exponents of the matrix coefficients of a representation with the parameter $\nu$. The quotient $J(P, \sigma, \nu)$ is the image of a standard intertwining operator, which converges because $\operatorname{Re} \nu$ lies in the positive chamber. If $G$ is semisimple and $P = G$, then $A$ is trivial, so $\nu = 0$, and $J(G, \sigma, 0) = \sigma$ runs over the tempered representations themselves. Example: let $G = \mathrm{SL}_2(\R)$ and let $P$ be the upper triangular matrices. The constant functions form a subrepresentation of the principal series $I(\mathrm{triv}, -\rho)$, and dually the trivial representation is a quotient of $I(\mathrm{triv}, \rho)$. So $J(P, \mathrm{triv}, \rho)$ is the trivial representation. The positivity of $\operatorname{Re} \nu$ is needed: on the unitary axis $\operatorname{Re} \nu = 0$ the induced representation can be reducible with several irreducible quotients, so uniqueness fails.

Going deeper

For a student

Langlands proved the classification for real groups; the analogous result for $p$-adic groups is due to Borel–Wallach and Silberger. Tempered representations are in turn built from discrete series of Levi subgroups, so the whole picture rests on Harish-Chandra's discrete series. The classification is the starting point of the local Langlands correspondence for real groups, which attaches $L$-parameters to the triples $(P, \sigma, \nu)$.