ABV Packets for G₂

The packet modes use Singular in your browser. After computing, click View Singular log to open Singular computation log — submitted script and raw output below the packet table. It shows the exact equations and commands submitted for that packet calculation, together with Singular's response. This log is available in both parameter modes. Download Singular script saves that program as a .sing file. JSON export includes the entire execution record, quotient bases, point counts, and microlocal characters. The engine is downloaded once per page (about 47 MB); subsequent calculations reuse its compiled module. Cancel computation stops an active calculation.

Unramified parameters

The Unramified parameters mode treats unramified Langlands parameters for split $G_2(F)$, or equivalently unipotent representations. It uses the Vogan varieties classified by Cunningham–Fiori–Zhang, Sections 1–2. The Dihedral parameters mode also supports ramified Weil parameters, in both commuting root embeddings, as described below.

A parameter is specified by its infinitesimal parameter $\lambda$ and a nilpotent orbit in $V_\lambda$. Enter

$$(a_1,t_1),(a_2,t_2) \quad\text{for}\quad \lambda(\mathrm{Fr})=\widehat m(e^{2\pi it_1}q^{a_1},e^{2\pi it_2}q^{a_2}).$$

Exponents and phases are rational; phases are taken modulo $1$. This includes every half-integral exponent and every finite-order unramified unitary part. The app compares rational values exactly, including fractions with large denominators. The residue cardinality $q>1$ remains symbolic.

The torus coordinates and the $q$-eigenspace convention follow the reference: in the dual root system, $\alpha=\widehat\gamma_1$ is long and $\beta=\widehat\gamma_2$ is short, with

$$\alpha(\widehat m(x,y))=x^{-1}y^2, \qquad \beta(\widehat m(x,y))=xy^{-1}.$$

The app computes all root eigenvalues and applies Weyl reflections to identify one of the nine normal forms in Proposition 1.1. Choose the orbit in that normal form using the selector; the resulting parameter is labelled $\phi_0,\phi_{1a},\ldots,\phi_{8d}$ as in the paper. The chosen orbit and normalization are displayed with the result. In Case 8, orbit representatives are written as binary cubics in the model $\det^{-1}\otimes\mathrm{Sym}^3$.

Representation labels use normalized parabolic induction $I_{\gamma_i}$ and Langlands quotients $J_{\gamma_i}$ in the notation of Section 2.6. Here $\gamma_1$ is short and $\gamma_2$ is long in the original group. The unramified characters satisfy $\nu(\varpi)=q$ and $\theta_n(\varpi)=e^{2\pi i/n}$, following the paper's convention. $I_0(G_2[\zeta])$ denotes compact induction of the indicated finite cuspidal unipotent representation from $G_2(\mathcal O_F)$.

Dihedral parameters

This mode uses the commuting root pair $(\mathrm{SL}_{2,l}\times\mathrm{SL}_{2,s})/\mu_2\subset G_2(\mathbb C)$. Put an irreducible dihedral Weil representation $r$ of determinant $1$ in one factor and the torus $\operatorname{diag}(\mu\nu^s,\mu^{-1}\nu^{-s})$ in the other. Here $\nu(\mathrm{Fr})=q$ and $\mu$ is unramified with $\mu(\mathrm{Fr})=e^{2\pi it}$. The root selector names the commuting factor:

• Long root: $r$ lies in the short-root $\mathrm{SL}_2$.
• Short root: $r$ lies in the long-root $\mathrm{SL}_2$.

At $(s,t)=(1/2,0)$ these give the infinitesimal parameters of the corresponding root Arthur parameters. The long-root convention agrees with Alonso–He–Ray–Roset, Definition 4.1 and Theorem 4.10. Other rational exponents and phases are allowed; this is not limited to the Arthur point.

The tame local parameter option specifies the Weil representation completely by the residue cardinality $q$ and inertia angle $a$:

$$r(\sigma)=A=\operatorname{diag}(e^{2\pi ia},e^{-2\pi ia}),\qquad r(\mathrm{Fr})=J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.$$

The app checks that $q$ is a prime power, that $a$ has order $N>2$ prime to the residue characteristic, and that $(q+1)a$ is integral. Thus it verifies $JAJ^{-1}=A^q$, $J^2=-1$, and $\det r=1$. This is the dihedral induction from the unramified quadratic extension, with a conjugate-symplectic inducing character. Different primitive inertia angles remain part of the parameter.

For a supplied Weil quotient, the app also permits:

• Cyclic inertia: the same matrices $A,J$, allowing a wild $p$-part in $N$. The prime-to-$p$ part must still divide $q+1$; this necessary tame-quotient relation is checked.
• Binary dihedral inertia: inertia is generated by $A$ and $J$, with even $N\ge4$ and $J^2=A^{N/2}$. Frobenius is $\operatorname{diag}(e^{2\pi iu},e^{-2\pi iu})$; the app checks $2Nu\in\mathbb Z$ so that it normalizes inertia. It also checks that the inertia image has a normal $p$-Sylow with cyclic tame quotient, and that Frobenius acts by $q$ on that quotient.

In these two supplied-quotient modes, the quotient and its map from the local Weil group are part of the input. The app verifies the matrix normalization relations and computes the resulting geometry; it does not construct a wild extension from a field polynomial or decide local realizability of an arbitrary finite quotient. The datum label retains the user's chosen $(K/F,\chi)$ or Weil representation. Packet members are specified by enhanced Langlands parameters $\pi(\phi,\eta)$ using this datum, rather than names of finite compact types.

Computing the ramified Vogan variety

The engine constructs all fourteen adjoint weight vectors using

$$\mathfrak g_2\big|_{\mathrm{SL}_{2,l}\times\mathrm{SL}_{2,s}} =\operatorname{Sym}^2(2_l)\oplus\operatorname{Sym}^2(2_s) \oplus\bigl(2_l\otimes\operatorname{Sym}^3(2_s)\bigr).$$

It imposes every inertia equation, diagonalizes Frobenius on the fixed space, and computes

$$V_\lambda=(\mathfrak g_2^{I_F})^{\operatorname{Ad}\lambda(\mathrm{Fr})=q}, \qquad \operatorname{Lie}H_\lambda=(\mathfrak g_2^{I_F})^{\operatorname{Ad}\lambda(\mathrm{Fr})=1}.$$

It also solves the group centralizer equations, retaining disconnected components. In the cyclic-inertia cases these reduce to an isometry equation $gBg^t=B$ in $\mathrm{GL}_2$, or in $\mathrm{SL}_3$ at the exceptional order-three inertia parameter. The symmetric and alternating forms distinguish $\mathrm O_2$, $\mathrm{SO}_3$, and $\mathrm{SL}_2$.

The resulting point or weighted-line geometry goes directly to the vanishing-cycle engine. For a line of weight $2$, the cover $z^2=x$ computes the nontrivial microlocal character; the closed packet contains one L-packet member and one coronal member. For a point, all enhancements of $\pi_0H_\lambda$ are retained. For the disconnected $\mathrm O_2$ point, the Aubert involution exchanges the two enhancements: the relative rank-one Jacquet formula gives $D(\pi_\pm)=\pi_\pm-(\pi_++\pi_-)=-\pi_\mp$.

Examples worth comparing:

• At $a=1/3$, $s=1/2$, $t=0$, both embeddings have the expected two-member dihedral Arthur packet at the zero orbit.
• At $a=1/3$, $s=1$, $t=1/4$, the long-root embedding has a weight-one line coming from the mixed tensor summand. Its two packets are singletons.
• At $a=1/3$, $s=0$, $t=1/4$, the short-root embedding has a point with $H_\lambda=\mathrm O_2$ and a two-member L-packet. The long-root embedding instead has connected $H_\lambda=\mathrm{SO}_3$ and a singleton packet.

The JSON export includes the complete Weil input, fourteen adjoint basis vectors, inertia-fixed Frobenius eigenspaces, group centralizer equations, and the runtime vanishing-cycle calculation. An independent optional audit, node --import tsx scripts/verify-g2-dihedral-adjoint.ts, checks the invariant spaces against exact $14\times14$ matrices using SymPy.

Reading a packet

An L-packet member has the input Langlands parameter. A coronal member has another Langlands parameter with the same infinitesimal parameter. The LLC enhancement belongs to the component group of the member's own parameter. The microlocal character belongs to the input parameter's group $A_\phi^{\mathrm{ABV}}$, which can be larger than $A_\phi$.

Membership is determined by a nonzero computed microlocal character. The expandable coefficient matrix includes the zero entries, so it also records exclusions. Microlocal rank is the character's dimension. For example, input (1,0),(1,0) and the zero-cubic orbit gives $\Pi^{\mathrm{ABV}}_{\phi_{8a}}$ with three members, one of rank $2$.

The app displays character values on every conjugacy class of the microlocal group. Here $\vartheta_n$ is the standard character of $\mu_n$, $\varepsilon$ is the sign character of $S_3$, and $\rho$ is its two-dimensional reflection representation. These are the unsigned microlocal characters $\langle\,\cdot\,,\pi\rangle$; the distribution in Equation (1) of the reference also includes $(-1)^{\dim C_\phi-\dim C_\pi}$.

The Aubert dual is displayed as an irreducible representation, omitting the overall sign of the Aubert operator. For the cubic case it uses Cunningham–Fiori–Zhang, Proposition 2.19. The link on the dual parameter opens its packet; shared URLs and JSON downloads retain the entire calculation, including the coefficient matrix.

Vanishing cycles from equations

The built-in resolutions and covers specify the IC objects. For each run, the app generates a finite collection of exact polynomial problems and submits them to Singular. It reads back standard bases, quotient monomial bases, and reduced projective point counts; these values enter the calculation of the microlocal ranks and characters. Published packet coefficients are regression fixtures only. Engine errors stop the calculation and do not trigger a fallback to the direct implementation.

For the point, line and plane models, it solves the support and conormal equations of the smooth orbit closures. The toric models also have Kummer local systems: the cover $z^n x_1=x_2$ and its cyclic projectors determine the vanishing cycles along each coordinate boundary, including their characters.

For binary cubics $f=au^3+bu^2v+cuv^2+dv^3$, it constructs the proper incidence maps

$$\pi_k:E_k=\{([l],f):l^k\mid f\}\longrightarrow V,\qquad k=1,2,3.$$

The small maps $\pi_1$ and $\pi_2$ construct the reflection local system and the discriminant IC sheaf. The semismall map $\pi_3$ constructs the triple-root IC sheaf after removing its exceptional delta summand. On transverse slices, polynomial substitutions give the cusp normalization $(c,d)=(-3z^2,-2z^3)$ and the double-root germ $d=-z^2$. The engine computes their Jacobian quotients and the action on vanishing cohomology, including the orientation factor.

At the zero cubic, the three critical points of $c\,\xi(l^3)$ are found by squarefree projective polynomial arithmetic. Their permutation representation, minus the exceptional constant summand, gives the rank-two microlocal stalk. Genericity is checked by solving the derivative equations exactly.

The sign local system is the alternating summand of the ordered-root cover. At the vertex its vanishing cycles come from

$$Z_\xi=\{B_\xi(l_1,l_2,l_3)=0\}\subset(\mathbb P^1)^3,$$

where $B_\xi$ is the polarization of the conormal cubic. The engine solves the fixed-locus equations for the commuting root-ordering and conormal $S_3$ actions. It computes their Euler characteristics by projective intersections and applies the alternating projector

$$\operatorname{tr}(g)=\frac16\sum_{h\in S_3} \operatorname{sgn}(h)\,\chi\bigl(Z_\xi^{(g,h)}\bigr).$$

The two boundary calculations use the discriminant double cover $w^2=q^2-p^3$, whose transverse Milnor fibre is $\mathbb C^*$, and the quadratic Kummer germ. These recover the microlocal characters, including monodromy, rather than just characteristic-cycle ranks.

Polynomial arithmetic takes place over $\mathbb Q$ or $\mathbb Q(\theta_3)$; finite Weil eigenvalues use exact rational phases. The expandable runtime-calculation section and JSON export include the equations, critical-point results, and fixed-locus Euler matrix. The resolution and cover descriptions follow Cunningham–Fiori–Zhang, Sections 3.5–3.10. The implementation is specialized to these G₂ geometric models; it does not accept arbitrary ideals or construct IC complexes for arbitrary singularities. Singular performs the polynomial quotient and critical-point computations; TypeScript performs exact substitutions, finite-character projectors, and the model-specific cohomological assembly. The resolutions, covers and Euler characteristic decompositions remain mathematical input built into the model. Representation names and Aubert duals use the established LLC classification.

The pinned WebAssembly engine, its licenses, provenance and source links are included under static/wasm/singular/; verify the engine files with node scripts/verify-abv-assets.mjs.