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7 votes
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105 views

Langlands correspondence of coverings of $\mathrm{SL}_2(\mathbb R)$ and modular forms with fractional weights

$\DeclareMathOperator\SL{SL}$Let $G \to \SL_2(\mathbb R)$ be a finite covering of degree $d \geq 2$. Then $G$ is a connected Lie group with semisimple Lie algebra $\mathfrak{g}=\mathfrak{sl}_2$ and ...
Zhiyu's user avatar
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6 votes
0 answers
190 views

Eigenvalues of spherical function on $\mathrm{SL}(2,\mathbb{R})$

Lie algebraically, the eigenvalue of the spherical function \begin{align*} \phi_{\lambda}(g)=\int_{K} e^{(i \lambda+\rho)(A(k g))} \mathrm{d} k \quad (g \in G,\,\lambda\in\mathfrak{a}^*) \end{align*} ...
user48713's user avatar
2 votes
0 answers
124 views

Modular transformation of affine characters of non-simply connected groups$.$

Consider an (untwisted) affine algebra corresponding to a compact and simply-connected Lie group $G$. Under a modular transformation, its characters transform as (cf. 9612078) $$ \chi_\mu\to\sum_{\nu\...
AccidentalFourierTransform's user avatar
22 votes
3 answers
2k views

Is SL(2,C)/SL(2,Z) a quasi-projective variety?

Consider the complex 3-fold $SL(2,\mathbb C)/SL(2,\mathbb Z)$ (just for clarity: note that $SL(2,\mathbb Z)$ acts without stabilizers, so this is a complex manifold, not a complex orbifold). Is $SL(...
John Pardon's user avatar
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