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The following quantity has come up in some work my collaborators and I are doing on equivariant D-modules, and in that particular context it seems to be very significant (i.e. it's the only "interesting" parameter in a certain naturally-defined family of D-modules). Has this quantity come up before?

Here are the details:

Let $G$ be a semisimple complex algebraic group, $\mathfrak{g}$ its Lie algebra, $\mathfrak{h}$ a Cartan subalgebra. Fix a choice of positive roots, and let $\rho$ be the Weyl vector (i.e. the half-sum of the positive roots). Let $W$ be an irreducible $G$-representation with highest weight $\mu$. The quantity in question is $$\frac{2\langle \rho, \mu\rangle}{\langle\mu,\mu\rangle},$$ where $\langle-,-\rangle$ is the bilinear form on $\operatorname{Hom}(\mathfrak{h},\mathbb{C})$ dual to (the restriction to $\mathfrak{h}$ of) the Killing form.

In our particular case, $W$ is $H^0(X,\mathcal{L})$, where $X$ is a projective homogeneous space for $G$, and $\mathcal{L}$ is a very ample $G$-equivariant line bundle on $X$.

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    $\begingroup$ In the usual root system notation (following e.g. en.wikipedia.org/wiki/Root_system#Definition), where $(\cdot, \cdot)$ is the Killing form, the quantity you are talking about would usually be denoted $\langle \rho, \mu \rangle$. Unless I'm confusing something... $\endgroup$
    – Sam Hopkins
    Commented Jul 26, 2022 at 15:24
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    $\begingroup$ It is related to Hilbert polynomial for the dimensions $H^0(X,\mathcal{L}^{\otimes n})$ as a polynomial in $n$. The top degree term in the polynomial is a homogeneous polynomial in $n$. If you divide through by this homogeneous polynomial, you get something like $1-(2\langle \rho,\mu\rangle/\langle \mu,\mu \rangle) (\text{dim}(X)/n)$ plus lower order terms. This follows from the Koll'ar -- Matsusaka version of "asymptotic Riemann-Roch". $\endgroup$
    – Jason Starr
    Commented Jul 26, 2022 at 15:25
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    $\begingroup$ @SamHopkins That is true, but usually those terms are only defined if the input weights are roots (they certainly could be defined for other weights, but they usually only come up for roots). $\endgroup$
    – Jason Starr
    Commented Jul 26, 2022 at 15:31
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    $\begingroup$ @jasonstarr morally, the Hilbert polynomial thing is where this quantity came from $\endgroup$
    – Avi Steiner
    Commented Jul 26, 2022 at 22:35
  • $\begingroup$ Those two terms in the Hilbert polynomial (the highest degree term together with the next highest degree term) also arise in several invariants considered in algebraic geometry: the "alpha invariant" and the Futaki invariant, for instance. $\endgroup$
    – Jason Starr
    Commented Jul 27, 2022 at 0:01

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