L2induced

L2induced, or L^2-induction, is a construction in the representation theory of locally compact groups used to build unitary representations of a group G from unitary representations of a closed subgroup H. It generalizes classical induction by using square-integrable sections over the homogeneous space G/H.

The standard setup begins with a locally compact group G, a closed subgroup H, and a unitary representation π of H on a Hilbert space V. The L^2-induced representation Ind_H^G^L2(π) acts on a Hilbert space of square-integrable sections of the associated vector bundle over G/H. Concretely, one considers measurable functions f: G → V satisfying f(gh) = π(h^{-1}) f(g) for all g ∈ G and h ∈ H, with finite norm ∥f∥^2 = ∫_{G/H} ⟨f(g), f(g)⟩ dμ(gH), where μ is a left-G-invariant measure on G/H. Functions are identified up to the natural H-adjoint relation, and G acts by left translation [g0 f](g) = f(g0^{-1} g). This yields a continuous unitary representation of G on the resulting Hilbert space.

L2-induction is related to, and often compared with, compact induction. If the homogeneous space G/H is compact, L^2-induction coincides with the usual compact induction. In general, L^2-induction interacts well with transitivity of induction and plays a role in Frobenius reciprocity for unitary representations.

Applications of L2induced representations appear in harmonic analysis on homogeneous spaces, the study of the unitary dual of G, and areas such as automorphic forms and the representation theory of p-adic and real Lie groups. It provides a framework for constructing and analyzing principal series and other induced representations.