Why a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$?

Question

It is said that a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$. Why a modular form is a highest weight vector of a discrete series summand of $L_2(SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R}))$? Could you explain this fact by some examples? Thank you very much.

Answer 1

The irreducible $G=SL_2(\mathbb R)$ representations occurring ("discretely") in $L^2$ of the arithmetic quotient must be irreducible repns of $G$. These are classified, in effect already by using asymptotics of ODEs, by Bargmann, Wigner, and others in that time. Namely, there are principal series (generically irreducible), and holomorphic and anti-holomorphic discrete series. (The latter two get stuck together if we allow signs...)

The holomorphic (and anti-holomorphic) irreds are subrepns of not-unitary principal series. In great generality, for reductive and semi-simple real Lie groups, unitary irreducibles are "admissible" (Harish-Chandra), and admissible irreducibles are subrepns of principal series (Casselman-Milicic).

It is useful that the only irreds with $K$-fixed vectors are the irreducible principal series, not the holomorphic or anti-holomorphic.

Given this classification, the practical question is to see what classical conditions ("waveform", "holomorphic", ... and why no others?) become when automorphic/modular forms on the domain upper-half-plane are converted into automorphic forms on the real Lie group $G$.

By some form of Schur's lemma, the Casimir operator is a scalar on irreducibles.

Waveforms on the domain become right-$K$-invariant Casimir eigenfunctions on the group, thus, necessarily generating principal series.

Holomorphic modular forms become right-$K$-equivariant Casimir eigenfunctions, and the holomorphic further translates (by a computation) into annihilation by a raising/lowering operator (depending on one's normalization).

Probably you want not so much an "example" but to carry out (or see carried out) some computation in coordinates certifying that holomorphic modular forms are converted into Casimir eigenfunctions and also annihilated by the raising/lowering operator. Indeed, this is a local issue, and not so much an issue about modular forms, although that's the way it seems to arise.