Representation Theory of SL2(R) - Admissible Representations

Admissible Representations

Using the fact that it is an eigenvector of the Casimir operator and has an eigenvector for H, it follows easily that any irreducible admissible representation is a subrepresentation of a parabolically induced representation. (This also is true for more general reductive Lie groups and is known as Casselman's subrepresentation theorem.) Thus the irreducible admissible representations of SL(2,R) can be found by decomposing the principal series representations I_ε,μ into irreducible components and determining the isomorphisms. We summarize the decompositions as follows:

• I_ε,μ is reducible if and only if μ is an integer and ε=−(−1)μ. If I_ε,μ is irreducible then it is isomorphic to I_ε,−μ.
• I_{−1, 0} splits as the direct sum I_ε,0 = D₊₀ + D₋₀ of two irreducible representations, called limit of discrete series representations. D₊₀ has a basis w_j for j≥1, and D_-0 has a basis w_j for j≤−1,
• If I_ε,μ is reducible with μ>0 (so ε=−(−1)μ) then it has a unique irreducible quotient which has finite dimension μ, and the kernel is the sum of two discrete series representations D_+μ + D_−μ. The representation D_μ has a basis w_μ+j for j≥1, and D_-μ has a basis w_−μ−j for j≤−1.
• If I_ε,μ is reducible with μ<0 (so ε=−(−1)μ) then it has a unique irreducible subrepresentation, which has finite dimension μ, and the quotient is the sum of two discrete series representations D_+μ + D_−μ.

This gives the following list of irreducible admissible representations:

• A finite dimensional representation of dimension μ for each positive integer μ, with central character −(−1)μ.
• Two limit of discrete series representations D₊₀, D₋₀, with μ=0 and non-trivial central character.
• Discrete series representations D_μ for μ a non-zero integer, with central character −(−1)μ.
• Two families of irreducible principal series representations I_ε,μ for ε≠−(−1)μ (where I_ε,μ is isomorphic to I_ε,−μ).