# Applying a simple involution to Hall-Littlewood polynomials

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.

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.