Brylinski–Deligne Covering Groups

• Brylinski–Deligne covering groups are central K2-extensions of reductive groups defined by three key invariants: a Weyl-invariant quadratic form, an abelian extension, and a compatible isomorphism.
• The dual group and L-group constructions for BD covers use modified root data to relate metaplectic-type extensions with Langlands parameters and genuine representations.
• These covers play a crucial role in harmonic analysis and the theory of automorphic forms by parametrizing ε-genuine representations via Whittaker models and local-global correspondence.

A Brylinski–Deligne (BD) covering group is a central extension of a connected reductive group $G$ by the Zariski sheaf $K_2$ of Quillen over a base scheme $S$ (e.g., a field, a Dedekind domain, a discrete valuation ring), regarded as sheaves of groups on the big Zariski site. This construction, foundationally developed by Brylinski and Deligne, produces (after pushout by the Hilbert symbol or a chosen character) the nonlinear "metaplectic-type" covers of algebraic and local/global groups pivotal to advances in the modern Langlands program, harmonic analysis, and the theory of automorphic forms.

1. Structural Foundation and Classification

BD covering groups arise concretely as central extensions

$1 \to K_2 \to G' \to G \to 1,$

in the category of sheaves of groups on $S_{\mathrm{Zar}}$. The main theorem of Brylinski–Deligne establishes an equivalence of Picard categories between such central extensions and a category $\mathrm{BD}_S(G,T)$ parameterized by three invariants:

• (a) A Weyl- and Galois-invariant quadratic form $Q\in H^0(S_{\mathrm{et}},\operatorname{Sym}^2 X)^W$ on the cocharacter lattice $Y$ of a maximal torus $T \subset G$;
• (b) A central extension of abelian sheaves

$1 \to \mathbb{G}_m \to D \to Y \to 1,$

whose commutator pairing recovers $(-1)^{B_Q(y_1,y_2)}$ with $B_Q(y_1,y_2) = Q(y_1 + y_2) - Q(y_1) - Q(y_2)$;

• (c) An isomorphism $f$ giving compatible identification on the simply-connected derived cover.

This classification is effective both over fields, where all extensions and invariants may be described using Galois modules, and over suitable rings (e.g., Dedekind domains, complete DVRs), provided mild assumptions such as the validity of Gersten's conjecture in weight two. The category of central $K_2$-extensions is a strictly commutative Picard category, with the Baer sum reflecting addition of invariants (Weissman, 2014).

2. Dual and L-Group Construction

Given a BD datum $(Q,D,f)$ for a central extension $\widetilde{G}$ of $G$ by $K_2$, one associates a modified root datum. For each simple coroot $\alpha^\vee$, define $n_\alpha = n/\gcd(n,Q(\alpha^\vee))$ and set: $$Y_{Q,n} = \{ y \in Y : B_Q(y, y') \in n \mathbb{Z},\;\forall y' \in Y \}, \quad \tilde{\alpha}^\vee = n_\alpha\,\alpha^\vee,$$ yielding a root datum $(X_{Q,n},Y_{Q,n})$ whose associated pinned complex reductive group is the "dual group" $\widehat{G}$ of the cover, with maximal torus $$\widehat{T} = \operatorname{Hom}(Y_{Q,n},\mathbb{C}^\times)$$ and center $$\widehat{Z} = \operatorname{Hom}(Y_{Q,n}/Y_{Q,n}^{sc},\mathbb{C}^\times)$$ (Weissman, 2015).

The L-group ${}^{\widetilde{G}}$ is constructed as an extension

$$1 \to \widehat{G} \to {}^{\widetilde{G}} \to \operatorname{Gal}(\bar{F}/F) \to 1,$$

built by the Baer sum of two key "twists":

• A "metaGalois extension" (the first twist), which captures the nontriviality of the Hilbert symbol on the Galois group;
• A "gerbe extension" (the second twist), reflecting the Kummer gerbe structure of the torsor of splittings of the central extension $D$ and the Whittaker datum.

Functoriality holds: well-aligned homomorphisms of BD covers (sending maximal tori, Borels, and respecting lattices as specified) induce natural transformations of L-groups, canonically up to unique isomorphism (Weissman, 2015).

3. Representation Theory and the Local-Global Langlands Program

The pushout of the $K_2$-extension $\widetilde{G}$ to $\mu_n$ using the Hilbert symbol produces a topological central extension

$1 \to \mu_n \to \widetilde{G}(F) \to G(F) \to 1.$

A central innovation is the parameterization of $\epsilon$-genuine irreducible admissible representations of $\widetilde{G}(F)$ in terms of ${}^{\widetilde{G}}$-valued Langlands parameters—continuous homomorphisms $\phi: W_F \to {}^{\widetilde{G}}$, up to $\widehat{G}$-conjugacy. This correspondence is fully established for covers of tori, for unramified representations, and for real double covers (Weissman, 2015, Gan et al., 2014, Weissman, 2011).

For tori, the Stone-von Neumann theorem links genuine characters of the center $Z(\widetilde{T})$ to splittings $W_F\to{}^{\widetilde{T}}$, yielding explicit parameterization of genuine irreducibles via $H^1(W_F, \widehat{T}^\#)$ and packet structure governed by Arthur's component group $S$ (Nakata, 27 Mar 2025). In the case of unramified principal series, the Satake isomorphism for covers produces a bijection between genuine unramified representations and $W$-conjugacy classes in ${}^{\widetilde{G}}$.

For $p$-adic and real groups, the situation is more delicate: the L-packets can contain multiple nonisomorphic genuine representations due to the nonuniqueness of Whittaker models and the finer structure of Arthur's $S$-group.

4. Harmonic Analysis and Whittaker Models

In linear groups, Whittaker models are unique for generic representations, but for BD covers, their dimension is governed by the arithmetic of $(Q,n)$ and fails to be one in general. Explicit formulas appear:

• The space of Whittaker functionals on a principal series $I(\chi)$ has dimension $|Y/Y_{Q,n}|$, reflecting the failure of the Gelfand–Kazhdan involution for covers (Gao et al., 2019, Gao et al., 2017).
• For theta representations attached to exceptional genuine characters, the dimension of the space of Whittaker functionals can be bounded by the number of free $W$-orbits on $Y$ modulo $Y_{Q,n}$ and can often be computed combinatorially (Gao, 2016).

The Casselman–Shalika formula is generalized to the covering setting, incorporating corrections via Gauss sums and interference from the covering cocycle (Cai, 2019). The dimension of Whittaker models is a critical invariant relevant to the construction of local and global $L$-functions via doubling and Rankin–Selberg integrals (Cai, 2019, Gao, 2014).

5. Harmonic Analysis on Parahoric and Levi Subgroups

BD covers naturally restrict to central extensions of parahoric subgroups and their reductive quotients, encoded by combinatorial invariants (such as functions $\kappa_x$ in the Bruhat–Tits building) that control the splitting of the cover (Weissman, 2010). For classical groups (symplectic, orthogonal), recent work shows that all Levi subgroups are decomposed (i.e., their preimages in the cover split as Baer sums) except in specific exceptional cases (e.g., split $\mathrm{SO}(4)$, certain GSpin and unitary groups). This decomposability is crucial for the extension of Bernstein decomposition and affine Hecke algebra formalism to the representation theory of BD covers (Li, 29 Sep 2025).

6. L-functions, Automorphic Forms, and the Global Theory

Automorphic partial $L$-functions for genuine automorphic representations of BD covers are defined by composing attached local parameters with algebraic representations of ${}^{\widetilde{G}}$, exactly as in the linear case. The constant terms of Eisenstein series and the relevant Gindikin–Karpelevich formula for BD extensions are expressible in terms of Langlands–Shahidi $L$-functions for the adjoint action of the L-group (Gao, 2014). The twisted doubling method and Rankin–Selberg theory extend to the covering case, enabling the construction of generalized Euler products (Cai, 2019).

7. Recent Developments and Functoriality

Progress in explicit structure theory includes:

• Computation of Arthur's $S$-groups and their role in the packet structure of genuine representations for certain BD covers of tori, with formulas depending on dual lattices and inertia action (Nakata, 27 Mar 2025);
• Development of "epipelagic" $L$-packets and stable conjugacy in BD covers, especially for symplectic groups, reconciling with established endoscopic and theta correspondence frameworks (Li, 2017).

Functoriality is built into the BD formalism at the categorical level through "well-aligned" homomorphisms, ensuring that the dual group and L-group construction is compatible with z-extensions, Levi subgroups, and various forms of Langlands functoriality, extending naturally to the covering context (Weissman, 2015, Gan et al., 2014).

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References

• "Covering groups and their integral models" (Weissman, 2014)
• "L-groups and parameters for covering groups" (Weissman, 2015)
• "The L-group of a covering group" (Weissman, 2015)
• "Managing Metaplectiphobia: Covering p-adic groups" (Weissman, 2010)
• "Stable conjugacy and epipelagic L-packets for Brylinski-Deligne covers of Sp(2n)" (Li, 2017)
• "Decomposed Levi subgroups in BD-covers of classical groups" (Li, 29 Sep 2025)
• "Distinguished theta representations for certain covering groups" (Gao, 2016)
• "Whittaker models for depth zero representations of covering groups" (Gao et al., 2017)
• "Local coefficients and gamma factors for principal series of covering groups" (Gao et al., 2019)
• "The Langlands-Weissman Program for Brylinski-Deligne extensions" (Gan et al., 2014)
• "Arthur's groups $S$ in local Langlands correspondence for certain covering groups of algebraic tori" (Nakata, 27 Mar 2025)
• "The Gindikin-Karpelevich Formula and Constant Terms of Eisenstein Series for Brylinski-Deligne Extensions" (Gao, 2014)
• "Unramified Whittaker functions for certain Brylinski-Deligne covering groups" (Cai, 2019)
• "Twisted doubling integrals for Brylinski-Deligne extensions of classical groups" (Cai, 2019)
• "Split metaplectic groups and their L-groups" (Weissman, 2011)