---
title: 'Quasi-Characters: Relaxed Notions Across Mathematics'
url: https://www.emergentmind.com/topics/quasi-characters
type: topic
---

# Quasi-Characters: Relaxed Notions Across Mathematics

“Quasi-character” is not a single standardized notion across mathematics and mathematical physics. In current usage it can denote at least six distinct constructions: a non-vanishing condition for irreducible characters of finite groups on \(p\)-regular elements, a homogeneity condition for restrictions of irreducible characters to normal subgroups, a root-theoretic condition on characters and cocharacters of connected reductive groups, a basis of invariant representative functions on \(G^N\) modulo diagonal conjugation, vector-valued modular functions arising from modular linear differential equations in vertex-operator and rational conformal field theory, and continuous homomorphisms from a locally compact abelian group to \(\mathbf C^\times\) in automorphic settings [2009.13412], [1708.07316], [2007.04855], [2208.09037], [1607.02910]. Several of these usages are explicitly noted to be unrelated despite the shared terminology [1708.07316], [2305.18574], [2207.01564].

## 1. Terminological scope and recurrent patterns

The common linguistic feature of the term is that it designates an object that relaxes a stricter notion while retaining part of its structure. In finite group character theory, “quasi \(p\)-Steinberg” keeps only the non-vanishing on \(p\)-regular elements, without requiring the exact Steinberg value formula \(\theta(x)=\pm |C_G(x)|_p\) [2009.13412]. In the theory of reductive groups, “quasi-constant” weakens “minuscule” by allowing constant absolute pairing \(m>1\) on a Weyl–Galois orbit rather than forcing values in \(\{-1,0,1\}\) [1708.07316]. In RCFT and VOA theory, quasi-characters are vector-valued modular functions with integral \(q\)-series coefficients but without the positivity required of admissible characters [1810.09472], [2507.07170]. In the automorphic setting of \(GL_2\) Eisenstein series, a quasi-character is simply a continuous homomorphism \(G\to \mathbf C^\times\), with unitary characters forming a distinguished subclass [1607.02910].

A concise comparison is useful.

| Domain | Meaning of “quasi-character” | Core relaxation |
|---|---|---|
| Finite groups | quasi \(p\)-Steinberg irreducible character | non-vanishing on \(p\)-regular elements only |
| Finite groups | quasi-primitive irreducible character | homogeneous restriction to every normal subgroup |
| Reductive groups | quasi-constant character/cocharacter | orbitwise values in \(\{0,\pm m_O\}\) |
| Compact Lie groups on \(G^N\) | invariant representative function basis element | generalizes ordinary characters from \(N=1\) |
| VOA/RCFT | VVMF with integral but not necessarily positive \(q\)-series | drops admissibility/positivity |
| Automorphic \(GL_2\) setting | continuous homomorphism to \(\mathbf C^\times\) | includes non-unitary twists |

This multiplicity of meanings is not accidental. Several sources explicitly warn that the same word is used differently in different subfields. The paper on quasi-constant characters states that its notion is unrelated to analytic or Harish-Chandra-style usages [1708.07316]. The note on quasi-primitive irreducible characters uses “quasi-characters” only for quasi-primitive characters in finite group theory [2305.18574]. The classification of quasi \(p\)-Steinberg characters of complex reflection groups likewise remarks that this usage is unrelated to quasi-characters in the Harish-Chandra or Arthur sense [2207.01564].

## 2. Finite-group character theory: quasi \(p\)-Steinberg and quasi-primitive characters

For a finite group \(G\) and a prime \(p\mid |G|\), an element \(g\in G\) is \(p\)-regular if \(p\nmid |g|\). An irreducible character \(\chi\in \mathrm{Irr}(G)\) is called quasi \(p\)-Steinberg if
\[
\chi(g)\neq 0 \qquad \text{for all } g\in G \text{ with } p\nmid |g|.
\]
This notion generalizes the classical Steinberg character by retaining only the non-vanishing on \(p\)-regular elements [2009.13412]. For symmetric and alternating groups, \(p\)-regularity is read off from cycle type: a class is \(p\)-regular iff no cycle length is divisible by \(p\) [2009.13412].

The classification for \(S_n\), \(A_n\), and their double covers is extremely rigid. For \(S_n\), non-linear quasi \(p\)-Steinberg characters exist only for \(n\le 8\), and the complete list is:
- \(n=3\): \((2,1)\) for \(p=2\).
- \(n=4\): \((2,2)\) for \(p=2\); \((3,1)\), \((2,1,1)\) for \(p=3\).
- \(n=5\): \((4,1)\), \((2,1,1,1)\) for \(p=2\); \((3,2)\), \((2,2,1)\) for \(p=5\).
- \(n=6\): \((3,2,1)\) for \(p=2\); \((4,2)\), \((2,2,1,1)\) for \(p=3\).
- \(n=8\): \((5,2,1)\), \((3,2,1,1,1)\) for \(p=2\).
For \(n\ge 9\), every non-linear irreducible character has a zero on some \(p\)-regular class [2009.13412].

For \(A_n\), the corresponding bound is \(n\le 9\). The list of non-linear quasi \(p\)-Steinberg characters is:
- \(n=3\): \((2,1)\) for \(p=3\).
- \(n=4\): \((2,2)\) for \(p=2\); \((3,1)\), \((2,2)\) for \(p=3\).
- \(n=5\): \((4,1)\) for \(p=2\); \((3,1,1)\) for \(p=3\); \((3,2)\) for \(p=5\).
- \(n=6\): \((3,2,1)\) for \(p=2\); \((4,2)\) for \(p=3\); \((5,1)\), \((3,3)\) for \(p=5\).
- \(n=8\): \((5,2,1)\) for \(p=2\).
- \(n=9\): \((7,2)\) for \(p=3\) [2009.13412].

For Schur double covers \(\widetilde S_n\) and \(\widetilde A_n\), ordinary characters inflated from \(S_n\) or \(A_n\) preserve the quasi \(p\)-Steinberg property, so the genuinely new issue is spin characters. The classification is strikingly sharp: no spin character is quasi \(p\)-Steinberg for odd \(p\), while for \(p=2\) the quasi \(2\)-Steinberg spin characters are exactly those indexed by strict partitions
\[
\lambda=(n),\ (3,1),\ (3,2),\ (3,2,1),\ (5,1),\ (5,2,1)
\]
[2009.13412].

The same non-vanishing problem was extended to complex reflection groups \(G(r,q,n)\). There the quasi \(p\)-Steinberg condition is defined identically, and the classification reduces most cases to the symmetric-group classification plus two low-degree families coming from restriction phenomena. For \(G(r,1,n)\), the only possibilities are certain \(r\)-partitions of shape \(\widehat\lambda^{\,j}\) corresponding to the \(S_n\) list above, together with three small mixed-shape families \(\widehat\lambda^{\,j,k}\) for \(n=2,3,4\). For \(G(r,q,n)\), all non-linear quasi \(p\)-Steinberg characters arise either by restriction of those \(G(r,1,n)\) cases or from two extra low-degree families: \(n=3\), \(r,q\) multiples of \(3\), giving quasi \(2\)-Steinberg constituents of degree \(2\); and \(n=4\), \(r,q\) even, giving quasi \(3\)-Steinberg constituents of degree \(3\). In particular, for \(n\ge 9\), no non-linear quasi \(p\)-Steinberg characters exist in \(G(r,q,n)\) [2207.01564].

A distinct finite-group usage is “quasi-primitive irreducible character.” An irreducible \(\chi\in \mathrm{Irr}(G)\) is quasi-primitive if for every normal subgroup \(N\triangleleft G\), the restriction \(\chi|_N\) is homogeneous:
\[
\chi|_N=e\cdot \theta
\]
for some \(e\ge 1\) and some \(\theta\in \mathrm{Irr}(N)\) [2305.18574]. Primitive characters are always quasi-primitive, but the converse need not hold in general [2305.18574]. The principal counting result is orbit-theoretic: both the number of primitive irreducible characters and the number of quasi-primitive irreducible characters are divisible by \(|G:G'|\), where \(G'\) is the derived subgroup [2305.18574]. The proof uses the multiplicative action of \(\mathrm{Irr}(G/G')\) on \(\mathrm{Irr}(G)\) and the fact that for quasi-primitive \(\chi\), the restriction \(\chi|_{G'}\) is irreducible, so the action is semiregular [2305.18574].

Methodologically, the finite-group classifications rely on explicit character formulas. For symmetric groups, the hook-length formula gives degrees and the Murnaghan–Nakayama rule supplies vanishing criteria. In the form used in the classification,
\[
\chi^\lambda(\alpha)=\sum_{\nu} (-1)^{\mathrm{ht}(\nu)-1}\chi^{\lambda\setminus \nu}(\alpha\setminus \alpha_i),
\]
and the practical consequences are that absence of an appropriate rim hook forces vanishing on a conjugacy class [2009.13412]. For spin characters of double covers, Schur’s vanishing theorem and Morris recursion play the analogous role [2009.13412]. This suggests that, in these settings, quasi-character phenomena are controlled less by abstract block theory than by fine combinatorics of Young and shifted diagrams.

## 3. Quasi-constant characters and cocharacters of reductive groups

In the theory of connected reductive groups, the relevant notion is not a class function but a weight in the root datum. Let \(G\) be a connected reductive group over a field \(k\), \(T\) a maximal torus, and \((X^*(T),\Phi;X_*(T),\Phi^\vee)\) its root datum. A character \(\chi\in X^*(T)\) is quasi-constant if for every root \(\alpha\in \Phi\) with \(\langle \chi,\alpha^\vee\rangle\neq 0\) and every \(\sigma\in W\rtimes \mathrm{Gal}(\bar k/k)\),
\[
\frac{\langle \chi,\sigma\alpha^\vee\rangle}{\langle \chi,\alpha^\vee\rangle}\in \{-1,0,1\}.
\]
Equivalently, for each Weyl–Galois orbit \(O\subset \Phi^\vee\), the multiset of pairings \(\{\langle \chi,\beta^\vee\rangle:\beta^\vee\in O\}\) is contained in \(\{0,\pm m_O\}\) for some integer \(m_O\ge 0\) [1708.07316]. The dual definition for cocharacters exchanges roots and coroots [1708.07316].

This notion interpolates between minuscule and cominuscule. Minuscule means \(\langle \chi,\alpha^\vee\rangle\in\{-1,0,1\}\) for all roots; quasi-constant allows a larger constant absolute value on an orbit. In the simple case, the nonzero quasi-constant characters are exactly the integer multiples of minuscule or cominuscule fundamental weights [1708.07316]. In simply-laced types, cominuscule and minuscule coincide, so quasi-constant means “multiple of a minuscule fundamental weight” [1708.07316].

The type-by-type classification is explicit. For simple, simply connected or adjoint groups over an algebraically closed field:
- \(A_n\): all fundamental weights are minuscule, so quasi-constant characters are multiples of any \(\eta(\alpha)\).
- \(D_n\): quasi-constant characters are multiples of \(\eta(\alpha_1)\), \(\eta(\alpha_{n-1})\), \(\eta(\alpha_n)\).
- \(E_6\): multiples of \(\eta(\alpha_1)\), \(\eta(\alpha_6)\).
- \(E_7\): multiples of \(\eta(\alpha_7)\).
- \(E_8\), \(F_4\), \(G_2\): no nontrivial quasi-constant characters.
- \(B_n\): multiples of \(\eta(\alpha_1)\) and \(\eta(\alpha_n)\).
- \(C_n\): multiples of \(\eta(\alpha_1)\) and \(\eta(\alpha_n)\) [1708.07316].

The general reductive classification reduces to \(k\)-simple factors. A character is quasi-constant iff its pullback to every \(k\)-simple factor of the simply connected cover of \(G^{\mathrm{der}}\) is quasi-constant, and on each absolutely simple factor the nontrivial components are all minuscule or all cominuscule, with a common integer scalar \(m\) across Galois-conjugate factors [1708.07316].

A major application concerns Shimura varieties. For a symplectic embedding \(\psi:(G,X)\hookrightarrow (GSp(2g),X_g)\), the character \(\eta_\omega(\psi)\) of the Hodge line bundle is quasi-constant for every such embedding [1708.07316]. When \(G^{ad}\) is \(\mathbf Q\)-simple, the positive ray of the Hodge line bundle in \(\mathrm{Pic}(Sh_K(G,X))_\mathbf Q\) is therefore independent of the symplectic embedding [1708.07316]. Another application is to \(G\)-zips: if \(G\) is over \(\mathbf F_p\) and \(\mu\in X_*(G)\) is quasi-constant, the duality construction yields a quasi-constant character \(\mu^*\in X^*(L)\) such that \(-\mu^*\) is a Hasse generator for \(G\text{-Zip}^\mu\), implying uniform principal purity for the zip stratification at all primes \(p\) [1708.07316].

The paper also formulates a canonical duality on rays. For semisimple \(G\), a \(\Delta\)-dominant quasi-constant ray in \(X_*(T)_\mathbf Q\) determines a \(\Delta\)-dominant quasi-constant ray in \(X^*(T)_\mathbf Q\) by replacing the unique excluded simple root in each Levi factor by the corresponding fundamental weight [1708.07316]. This suggests a structural symmetry between Hodge cocharacters and line-bundle characters that is sharper than mere root-datum duality.

## 4. Quasicharacters as invariant functions on \(G^N\)

For a compact Lie group \(G\), another usage concerns invariant representative functions on \(G^N\) under diagonal conjugation. Let \(R=A(G^N)^G\), where \(A(G^N)\) is the algebra of representative functions and \(G\) acts by
\[
h\cdot (g_1,\dots,g_N)=(h g_1 h^{-1},\dots,h g_N h^{-1}).
\]
Fix irreducible unitary representations \((H_\lambda,D^{(\lambda)})\), write \(\Lambda=(\lambda_1,\dots,\lambda_N)\), and decompose the diagonal restriction \(D_\Delta^{(\Lambda)}\) into isotypical components. A reduction scheme chooses intertwiners
\[
p_{\lambda,\ell}:H_\Lambda\to H_\lambda,\qquad i_{\lambda,\ell'}:H_\lambda\to H_\Lambda
\]
with \(p_{\lambda,\ell}i_{\lambda,\ell'}=\delta_{\ell,\ell'}\mathrm{id}_{H_\lambda}\), and defines invariant representative functions
\[
\chi^{(\lambda)}_{\ell\ell'}(g_1,\dots,g_N)=\sqrt{\dim H_\lambda}\,\mathrm{tr}\!\big[D^{(\Lambda)}(g_1,\dots,g_N)\,\widehat A^{(\lambda)}_{\ell\ell'}\big].
\]
These are called quasicharacters [2007.04855].

For \(N=1\), quasicharacters reduce to ordinary irreducible characters \(\chi^{(\lambda)}(g)=\mathrm{tr}\, D^{(\lambda)}(g)\) [2007.04855]. For general \(N\), they depend on a reduction scheme, equivalently on a rooted binary tree \(\mathcal T\) encoding the successive Clebsch–Gordan reductions. In the tree language,
\[
\chi^{(\mathcal T)}_{aa'}(g_1,\dots,g_N)=\sum_{\mu\in w(\lambda)} \langle \mathcal T;a,\mu\mid D_\Delta^{(\Lambda)}(g_1,\dots,g_N)\mid \mathcal T;a',\mu\rangle,
\]
or, in index notation,
\[
\chi_{\mathcal T,\boldsymbol\lambda,\iota}(g_1,\dots,g_N)
=\sum_{m_i,n_i}\Big[\prod_{i=1}^N D^{(\lambda_i)}_{m_i n_i}(g_i)\Big]\mathcal C_\mathcal T^{\{m_i,n_i\}(\iota)}.
\]
Thus the quasicharacter is a contraction of \(N\) matrix elements along a fixed intertwiner pattern [2007.04855].

These functions form an orthogonal basis of \(L^2(G^N)^G\). Their product closes in the same basis:
\[
\chi_a\cdot \chi_b=\sum_c C_{ab}{}^c\,\chi_c,
\]
and the structure constants are expressed in terms of recoupling coefficients. In the tree-based formulation,
\[
\chi^{(\mathcal T)}_{a_1 a_1'}\chi^{(\mathcal T)}_{a_2 a_2'}
=\sum_{a_3,a_3',k,k'} R(\mathcal T)_{a_3,(a_1 a_2;k)}\,R(\mathcal T)_{a_3',(a_1' a_2';k')}\,\chi^{(\mathcal T)}_{a_3 a_3'},
\]
so the multiplication law is entirely controlled by recoupling theory [2007.04855]. The recoupling coefficients themselves factor into products over primitive \(9j\)-type quantities attached to the internal nodes of the tree [2007.04855].

For \(G=SU(2)\), the whole construction becomes angular momentum theory. Irreducibles are labelled by spins \(j\in \{0,\tfrac12,1,\dots\}\), multiplicities in two-fold tensor products are \(0\) or \(1\), and recouplings are Racah–Wigner coefficients. The quasicharacters become sums of diagonal matrix elements over magnetic quantum numbers, their norms are explicit in terms of \(d_j=2j+1\), and the \(9\mathfrak X\)-symbols reduce to Wigner \(9j\) symbols up to dimension factors [2007.04855].

The main motivation is Hamiltonian lattice gauge theory. With a maximal tree chosen in the lattice, the physical Hilbert space is \(L^2(G^N)^G\), and bi-invariant operators such as Casimirs, orbit-type relations, and the Kogut–Susskind Hamiltonian can be represented in the quasicharacter basis. Multiplication by an invariant representative function \(r\in R\) is reduced to finite-dimensional linear algebra using the structure constants \(C_{ab}{}^c\), while differential operators act diagonally through Casimir eigenvalues [2007.04855]. The paper works out explicit examples for \(SU(2)\) and \(SU(3)\), including orbit-type relations and sparse matrix elements [2007.04855].

Conceptually, these quasicharacters are close to spin networks. The paper states that for a single site with \(N\) incident edges, quasicharacters coincide with spin-network evaluations on a star graph reduced by the diagonal action; the difference lies in normalization conventions and in the explicit algebraic product law within \(R\) [2007.04855].

## 5. Quasi-characters in VOA and RCFT

In conformal field theory and vertex-algebra theory, “quasi-character” again has a different meaning. Here the fundamental objects are genus-one trace functions
\[
\mathrm{ch}_M(\tau)=\mathrm{Tr}_M\bigl(q^{L_0-\frac c{24}}\bigr),\qquad q=e^{2\pi i\tau},
\]
or Jacobi trace functions
\[
\mathrm{ch}_M(\tau,z)=\mathrm{Tr}_M\bigl(y^{J_0}q^{L_0-\frac c{24}}\bigr),\qquad y=e^{2\pi i z},
\]
and quasi-characters are \(q\)-series, often vector-valued, that solve modular linear differential equations, have integral Fourier coefficients after normalization, but fail positivity and therefore do not directly define characters of rational CFTs [2208.09037], [1810.09472].

The two-character case is the foundational example. A rank-2 vector-valued modular form has components
\[
\chi_i(\tau)=q^{\alpha_i}\sum_{n=0}^\infty a_{i,n}q^n,
\]
with \(\alpha_0=-c/24\), \(\alpha_1=-c/24+h\), and Wronskian index
\[
\ell=1-6h+\frac c2.
\]
For \(\ell=0\), the MLDE is the Mathur–Mukhi–Sen equation
\[
\bigl(D_\tau^2+\mu E_4(\tau)\bigr)\chi=0,\qquad \mu=-\frac{c(c+4)}{576}.
\]
Quasi-characters are weight-0 vector-valued modular forms solving such MLDEs with integral coefficients but not necessarily nonnegative ones [1810.09472], [2507.07170].

The classification of rank-2 quasi-characters at \(\ell=0\) organizes them into infinite families parameterized by
\[
c=24M+j,\qquad j\in \Bigl\{1,2,\frac{14}{5},4,\frac{26}{5},6,7\Bigr\},\qquad M\in \mathbf Z,
\]
with exponents
\[
\alpha_0=-M-\frac j{24},\qquad \alpha_1=M+\frac{j+4}{24},
\]
and family-dependent modular \(S\)-matrices that are independent of \(M\) [2507.07170]. The coefficient behavior is highly structured. For \(M>0\), the identity-component coefficients alternate in sign up to \(n=2M\), with sign \(({-1})^n\), and are strictly positive for all \(n>2M\); the non-identity component is strictly positive for every \(n\ge 0\). For \(M<0\), the identity component is strictly positive for all \(n\ge 0\), while the non-identity component alternates up to \(n<2|M|\), has \(a_{1,2|M|}<0\), and is strictly negative for all \(n>2|M|\) [2507.07170]. These results prove earlier conjectures about sign stabilization near \(n\approx c/12\) [2507.07170], sharpening the two-character quasi-character program initiated in the 2018 classification [1810.09472].

This sign structure is not merely descriptive. Quasi-characters form explicit bases from which admissible characters can be built by finite linear combinations within a fixed modular family:
\[
\chi_i(\tau)=\sum_{M=M_{\min}}^{M_{\max}} c_M\,\chi_i^{[24M+j]}(\tau).
\]
Such combinations have
\[
\ell=6(M_{\max}-M_{\min}),\qquad c=24M_{\max}+j,\qquad h=M_{\max}+M_{\min}+\frac{j+2}{12},
\]
and with suitable coefficients \(c_M\) can yield admissible character vectors with nonnegative integral coefficients [2507.07170]. Earlier work proved that in rank 2 all admissible characters of allowed \(\ell\) can be generated from quasi-characters in this way [1810.09472].

The three-character case is substantially more intricate. A third-order MLDE
\[
\bigl(D^3+\pi^2\mu_1 E_4 D+i\pi^3\mu_2 E_6\bigr)\chi(\tau)=0
\]
governs \((3,0)\) theories, and infinite families of three-character quasi-characters were conjectured and used to generate admissible characters of arbitrarily large Wronskian index [2002.01949]. A more recent development gives a universal hypergeometric description: all \((3,0)\) solutions can be written in terms of \({}_3F_2\), taking into account monodromy at the elliptic points. Starting from known \((3,0)\) and \((3,3)\) solutions, the matrix-MLDE formalism produces additional basis vectors with the same multiplier; these are typically quasi-characters. Integer linear combinations of them yield new admissible \((3,6)\), \((3,9)\), and higher-index solutions, and the admissible points appear as integer points in a polytope [2510.24248].

In affine \(\widehat{su(2)}\) current algebra at admissible fractional levels, quasi-characters arise from unflavoured even characters. The paper on fractional levels shows that outside three special classes—threshold levels, positive half-odd integer levels, and the isolated level \(-5/4\)—the resulting vectors are quasi-characters: vector-valued modular functions with integer \(q\)-series coefficients violating positivity [2208.09037]. At half-odd integer levels, the even characters map to differences of \(\widehat{su(2)}_{4N}\) characters, which explains why they do not define RCFTs despite often having positive coefficients to very high order [2208.09037].

The VOA perspective places these modular phenomena in a geometric framework. For a conformal vertex algebra \(V\), quasi-lisse and stably quasi-lisse conditions imply holonomicity of the sheaf of charged conformal blocks over the moduli of elliptic curves with line bundles. Under stable rationality, the space of flat sections is spanned by trace functions on irreducible stable modules [2605.29921]. The Jacobi-invariant connection satisfies Ward identities
\[
\frac{\partial}{\partial \alpha}S(u,\alpha,\tau)=S\bigl(\mathrm{Res}_t\,\wp_1(t,\tau)h(t)u,\alpha,\tau\bigr),
\]
\[
2\pi i\,\frac{\partial}{\partial \tau}S(u,\alpha,\tau)=S\bigl(\mathrm{Res}_t\,\wp_1(t,\tau)L(t)u,\alpha,\tau\bigr),
\]
and the resulting flat sections transform under the Jacobi group as vector-valued Jacobi forms [2605.29921]. In particular, for admissible affine vertex algebras \(L_k(\mathfrak g)\), the dimension of the space of conformal blocks equals the number of admissible weights at level \(k\), and the charged trace functions form a vector-valued Jacobi form of weight \(0\) and index
\[
\kappa=\frac{k\,h^\vee\,\dim(\mathfrak g)}{24}
\]
[2605.29921]. This gives a rigorous modular-geometric explanation for the appearance of quasi-character solutions in MLDE classifications.

## 6. Automorphic quasi-characters and broader perspective

In the automorphic setting of real-analytic Eisenstein series, a quasi-character is simply a continuous homomorphism from a locally compact abelian group to \(\mathbf C^\times\). For a locally compact abelian group \(G\),
\[
X(G)=\mathrm{Hom}_{\mathrm{cont}}(G,\mathbf C^\times)
\]
is the group of quasi-characters, while characters are those whose image lies in \(S^1\) [1607.02910]. For \(K\) a totally real field and \(G=K_\mathbf C^\times\), the monograph isolates a lattice of integral unitary characters
\[
\omega_p(z)=\prod_{i=1}^g \left(\frac{z_i}{|z_i|}\right)^{p_i},\qquad p\in \mathbf Z^g,
\]
and, relative to a finite-index subgroup \(V^+\subset O_K^\times(\infty)\), defines \(V^+\)-integral quasi-characters
\[
\chi(z)=\omega_p(z)\,\mathrm{Norm}(z)^w\,\eta_m(z),
\]
where \(p\in \mathbf Z^g\), \(w\in \mathbf Z\), and \(\eta_m\) is built from logarithms of a \(\mathbf Z\)-basis of \(V^+\) [1607.02910]. These are the monograph’s “multiplicative integral quasi-characters.”

They enter directly into the definition of twisted \(GL_2\) real-analytic Eisenstein series. Given lattices \(\mathfrak m,\mathfrak n\subset K\), a parameter matrix
\[
U=\begin{bmatrix}u_1 & v_1\\ u_2 & v_2\end{bmatrix}\in M_2(K),
\]
an integral weight \(p\in \mathbf Z^g\), and \(w\in \mathbf Z\), the Eisenstein series
\[
G_{(\mathfrak m,\mathfrak n)}^w(U,p; z,s)
\]
is defined by a lattice sum twisted by \(\omega_p\), \(\mathrm{Norm}^{-w}\), additive characters from \(u_1,u_2\), and the analytic factor \(|\mathrm{Norm}(\cdot)|^{-2s}\) [1607.02910]. The parameter matrix is acted on by \(GL_2(K)\) through an “upper right action,” and the associated Cartan involution \(U\mapsto U^*\) controls the functional equation [1607.02910].

The completed series satisfies a functional equation of the form
\[
\mathfrak G_{(\mathfrak m,\mathfrak n)}^0(U,p; z,s)
=
(-1)^{\mathrm{Tr}(p)} e^{2\pi i \mathrm{Tr}(\ell_U)}
\frac{\mathrm{cov}(\mathfrak n^*)}{\mathrm{cov}(\mathfrak m)}
\,
\mathfrak G_{(\mathfrak n^*,\mathfrak m^*)}^0(U^*,p; z,1-s),
\]
where \(\ell_U=u_1v_1+u_2v_2\) [1607.02910]. The Fourier expansion is explicit, with constant terms expressed in terms of partial zeta functions twisted by \(\omega_p\), and more general \(V^+\)-integral quasi-characters \(\omega_p\eta_m\mathrm{Norm}^w\) can be incorporated formally in the same framework [1607.02910].

Across all these subjects, the term “quasi-character” therefore functions as a marker of controlled generalization rather than a univocal definition. In finite groups it isolates non-vanishing or homogeneous-restriction phenomena; in reductive groups it encodes orbitwise rigidity of root pairings; in compact-group invariant theory it names a recoupling-adapted basis of gauge-invariant functions; in RCFT and VOA theory it identifies MLDE solutions that preserve modularity and integrality but not positivity; and in automorphic analysis it retains its classical meaning of a continuous homomorphism to \(\mathbf C^\times\). A plausible implication is that the persistence of the prefix “quasi-” reflects a shared methodological pattern: the relaxation of one axiom while preserving enough structure to retain classification, analytic continuation, or representation-theoretic control.

Source: https://www.emergentmind.com/topics/quasi-characters