Applying a simple involution to Hall-Littlewood polynomials

Question

Consider the Hall-Littlewood polynomial $$ P_\lambda(x_1,\ldots,x_n;t)=\sum_{\sigma\in S_n/S_n^\lambda}\sigma\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n}\prod\limits_{\lambda_i>\lambda_j}\dfrac{x_i-tx_j}{x_i-x_j}\right), $$ where $\lambda=(\lambda_1,\ldots,\lambda_n)$ is a partition and $S_n^\lambda$ is the stabilizer of $\lambda$. These give a $\mathbb{Z}[t]$-basis for the ring of symmetric functions (with coefficients in $\mathbb{Z}[t]$). In particular, if we apply the involution $$t\mapsto -t,$$ we get such a symmetric polynomial, $P_\lambda(x_1,\ldots,x_n;-t)$, which we can expand as a linear combination of Hall-Littlewood polynomials: ie. there are unique polynomials $h_{\lambda,\mu}(t)$ such that $$ P_\lambda(x_1,\ldots,x_n;-t)=\sum_{\mu}h_{\lambda,\mu}(t)P_\mu(x_1,\ldots,x_n;t). $$

Is there a known expression for the coefficients $h_{\lambda,\mu}(t)$?

A couple things to say: since $P_\lambda(x;0)=s_\lambda(x)$ is the Schur polynomial, we need $h_{\lambda,\mu}(0)=\delta_{\lambda,\mu}$. For example, when $n=2$, it is simple to compute that $$ P_{(\lambda_1,\lambda_2)}(x_1,x_2;-t)=P_{(\lambda_1,\lambda_2)}(x_1,x_2;t)+\sum_{k=1}^{[\lambda_1-\lambda_2/2]}(2t^k)P_{(\lambda_1-k,\lambda_2+k)}(x_1,x_2;t), $$ where $[n]$ is the floor function. This is clearly a root string, so I am hoping there is a known expression (say in terms of tableaux or something) in general.

A second, vaguer question is

is there is a theoretic interpretation to the involution $t\mapsto -t$ in relation to these polynomials and their generalizations? By theoretic, I am referring to the myriad ways in which HL polynomials appear (in terms of Hecke algebras or geometric representation theory).

This question arose from certain computations with $p$-adic groups, and this old question seems to indicate that there may be something interesting to say.

Answer 1

The transition matrix from the Schur functions to the HL symmetric functions is $K(t)$, the matrix of Kostka polynomials. This means that the transition matrix from $P(x;t)$ to $P(x;-t)$ is $K(t)^{-1}K(-t)$. This is upper-triangular with respect to the dominance partial order on partitions (or lower-triangular, depending on how you look at it), so $h_{\lambda\mu}(t) = 0$ unless $\lambda \succeq \mu$.

The character theory of $\text{GL}_n(\mathbb{F}_q)$ and $\text{U}_n(\mathbb{F}_{q^2})$ (the finite unitary group) can both be described in terms of symmetric functions, with the involution $q \leftrightarrow -q$ often relating the two structures. I don't know of any particularly meaningful interpretation of the HL symmetric functions in terms of characters of these groups, however.