ABV Packets for GLₙ

ABV Packet Calculator for $\mathrm{GL}_n$

Compute ABV packets for unramified, half-integral parameters of $\mathrm{GL}_n(F)$ over a non-archimedean local field. Enter $(d,s)$ for $\nu^s\otimes S_d$, where $\nu=|\cdot|$ and $\dim S_d=d$, or ascending segments $[a,b]$ for $\{\nu^a,\nu^{a+1},\ldots,\nu^b\}$.

Langlands parameter $(d,s)$ or multisegment $[a,b]$:

Entries per page:

Parameters and conventions

The input describes an unramified parameter $\phi=\bigoplus_i\nu^{s_i}\otimes S_{d_i}$, with $\nu=|\cdot|$, $s_i\in\tfrac12\mathbb Z$, and $\dim S_d=d$. Its group is $\mathrm{GL}_{\sum_i d_i}(F)$. The equivalent segment is $[s_i-(d_i-1)/2,s_i+(d_i-1)/2]$ with ascending endpoints. Repeated summands are allowed.

Representations are labelled by their Langlands-quotient multisegments. The Zelevinsky dual is computed using the Mœglin–Waldspurger algorithm. Independent connected strings of exponents give separate quiver factors; their packets and microlocal ranks combine by products.

How membership is determined

For each factor, the Vogan variety is the representation space of an equioriented type-A quiver. Path ranks describe its orbit closures. The two inequalities $C_\phi\leq C_\pi$ and $\widehat C_\phi\leq\widehat C_\pi$ are necessary conditions from Cunningham–Fiori–Kitt, Proposition 3.2.1. Passing them alone does not certify membership.

The app first uses applicable results, including the Arthur singleton theorem of Cunningham–Ray, Theorem 6.1, its common twists, and the explicitly computed Kashiwara–Saito packet of Cunningham–Fiori–Kitt, Theorem 3.1. The latter example has seven orbit-order candidates and exactly two members.

For remaining candidates, the app computes characteristic-cycle multiplicities:

Before computing polar varieties, it checks exact product and vanishing certificates. A zero arrow in the candidate splits its orbit closure into factors, even when the exponent support is connected. On a transverse slice, equivariant matrix blocks can also certify vanishing: a sufficiently large leaf group gives a Fourier-transform test, and a verified incidence bundle gives a test using generic linear forms. These tests check their geometric hypotheses on the actual slice equations. Cases that do not satisfy them continue through the calculation below.

1. Form an affine slice transverse to the input orbit, using its exact tangent space. Impose the candidate's path-rank equations on that slice.

2. Compute local Euler obstructions. After exact graph substitutions, homogeneous slices use the multidegrees of their conormal varieties. For positively weighted homogeneous slices, equivariant localization computes the Euler obstruction directly; see Aluffi–Mihalcea–Schürmann–Su, Theorem 4.2 and Weber, Theorem 20. The grading and the density of the chosen Jacobian chart are checked exactly. Other slices use polar multiplicities and the Lê–Teissier formula, with algebraically independent projection coefficients over $\mathbb Q$. Singular/WebAssembly computes saturations and standard bases.

3. Compute IC stalk Euler characteristics using ordinary Kazhdan–Lusztig polynomials for the maximal full-flag lifts of the Zelevinsky permutations; see Xu–Zhang.

4. Solve the triangular microlocal index formula

   $$(-1)^{\dim C_B}P_{A,B}(1) =\sum_{A\leq S\leq B}(-1)^{\dim C_S} m_{S,B}\operatorname{Eu}_{\overline{C_S}}(C_A).$$

   A positive $m_{A,B}$ certifies membership. A zero multiplicity excludes the candidate. The reported rank is the dimension of the generic microlocal stalk; its monodromy representation is not computed.

KL coefficients, conormal multidegrees, and Hilbert–Samuel multiplicities use arbitrary-precision integer arithmetic in TypeScript. Multidegrees are computed by recursively partitioning standard monomials, avoiding the full Hilbert numerator and Singular's fixed-width multiplicity output.

This gives a finite exact procedure for every parameter in the supported input class. Its running time and memory can grow very rapidly. A cancelled or failed calculation retains its certified results and labels the remaining candidates unresolved; it never interprets a resource failure as nonmembership.