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 $h\_{\lambda,\mu}(t)$?
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> 


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](https://mathoverflow.net/questions/190646/on-a-positivity-property-of-hall-littlewood-polynomials) seems to indicate that there may be something interesting to say.



Applying a simple involution to Hall-Littlewood polynomials
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.


