Apologies if this is not research level math (in that it concerns well-known stuff), but I am having trouble tracking down sources that explain the following. References would be very appreciated.

Let $S_n$ act on $R := \mathbb{C}[x_1,\ldots,x_n]$ (polynomial functions on $n$-dimensional affine space over $\mathbb{C}$), and let $R^{\mathrm{co}} := R/R^{S_n}_+$ denote the co-invariant algebra (where $R^{S_n}_+$ denotes the ideal generated by invariant polynomials with zero constant term).

As an $S_n$ representation $R^{\mathrm{co}}$ caries the regular representation $\mathbb{C}[S_n]$. Also, $R^{\mathrm{co}}$ inherits a grading from $R$. For a partition $\lambda\vdash n$, the fake degree polynomial $f^{\lambda}(q)$ is the Hilbert series of the $\lambda$-isotypic component of $R^{\mathrm{co}}$. (Actually I think we may need to divide by $\mathrm{dim}(V^{\lambda})$ to account for the fact that we have $\mathrm{dim}(V^{\lambda})$ copies of $V^{\lambda}$ in $\mathbb{C}[S_n]$.)

There is however, a completely different to define $f^{\lambda}(q)$. Namely, $f^{\lambda}(q)$ is the dimension of the irreducible unipotent representation of $GL_n(\mathbb{F}_q)$ indexed by $\lambda$.

It is my understanding that we can do the same thing in other types (i.e., for other Weyl groups and other classical groups over finite fields), but that outside of Type A the corresponding polynomials $f^{\lambda}(q)$ are not quite the dimensions of irreducible representations of these finite groups, but degrees of virtual characters or something like that, and this is where the name "fake degree" comes from.

Anyways, my question is: is there a simple explanation why the same polynomials occur in the Hilbert series for the coinvariant algebra of finite reflection groups and as degrees of irreducible representations of classical groups over finite fields? Is there an a priori heuristic that explains this?

  • $\begingroup$ Is your definition of "co-invariant algebra" the usual one? I am used to the term meaning the largest quotient (instead of the largest subobject) on which $\mathrm S_n$ acts trivially, namely, the quotient of $R$ by the ideal generated by $(1 - \sigma)r$ with $\sigma \in \mathrm S_n$ and $r \in R$. $\endgroup$ – LSpice Aug 19 '19 at 5:21
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    $\begingroup$ I know very little on this subject, but I think your question may be answered in Carter's Finite groups of Lie type book, Ch 10-12. The key factor, as far as I understand, in seeing this connection is the (non-trivial) isomorphism between $\mathbb C[W]$ and the Hecke algebra $\mathbb C[B\backslash G/B]$, and the notion of specialisation, which gives representation dimensions in the former case and fake degrees in the latter. $\endgroup$ – kneidell Aug 19 '19 at 8:46
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    $\begingroup$ @LSpice This is the standard meaning of "coinvariant algebra" in invariant theory. See eg math.uchicago.edu/~margalit/repthy/lecturesRT.pdf . It is unfortunate that it is not what happens if you apply the definition of "coinvariant" ncatlab.org/nlab/show/coinvariant to an algebra, but such is human language. $\endgroup$ – David E Speyer Aug 19 '19 at 10:05

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