In a paper by Yuji Tachikawa, I found a q-deformed "2d Yang-Mills paritition function for a cylinder". Here it is (adapted):

$$ Z(q, x_L, x_R) = \mu(q, x_L)^{-1/2} \langle x_L | \bigg\[ \sum_{R \in \mathrm{Irr}(G)} | R \rangle e^{- aC_2(R) } \langle R | \bigg\] |x_R \rangle \mu(q, x_R)^{-1/2}$$

Here's some stuff to help you interpret:

- $G$ is a compact lie group and the Irreducible representations should be indexed by the root lattice.
- conjugacy classes are indexed by elements of maximal torus $\vec{x} \in \mathbb{T}^n \subset G$
- $C_2(R)$ is the quadratic Casimir of the representation.
- In my notation, borrowed from quantum mechanics $\langle R|x \rangle = \chi_R(x), \langle x|R \rangle=\overline{\chi_R(x)}$.
- $\displaystyle \mu(q, X) = \exp \left[ \sum_{n=1}^\infty \frac{-2q^n}{1-q^n}\chi_{\mathrm{adj}}(x^n) \right]$
- The partition function depends on the area $a$ of the cylinder.

In fact, let's turn this into a statement about the Laplacian: The $q$-dependence is hidden:

$$ e^{- a \Delta} = \sum_{R \in \mathrm{Irr}(G)} | R \rangle e^{- aC_2(R) } \langle R | $$

Let's set the area to $0$. From the last line, we should get the identity matrix. However,

$$ \sum_{R \in \mathrm{Irr}(G)} \langle x_L | R \rangle \overline{ \langle x_R | R \rangle } = \mu(q, x_L) \delta(x_L = x_R)$$

This really looks like orthogonality of characters for compact groups, except the right side should be the identity.

**What are these characters $\langle x | R \rangle$ ?**

Originally, I wanted to ask about an analogue for finite $G$, but I don't even have a point of reference.