The Signed Adjoint of the Reflection on a Topological Group

Introduction

A reflection of a graded topological group is a signed conjugation that is an involution, and read on the group algebra it is a signed operator whose adjoint is again a reflection, the reflection carried by the inverse element. The questions the adjoint answers are the classical ones: when is a reflection self-adjoint, when is it orthogonal, and what happens when the carrying element fails the centrality that makes the operator an involution at all. The answers are clean in the reflection case — self-adjointness, unitarity and orthogonality all coincide and hold exactly when the square of the carrying element is central — and they separate in the degenerate case, which is the operator form of the degeneracy of the reflection correspondence.

The article assumes the signed form, the adjoints of the translations and the linear extension of the grade involution from The Adjoint of the Left Multiplication on a Topological Group; the adjoint of the signed sandwich, the inverse and the unitarity criterion from The Signed Adjoint Sandwich on a Topological Group; the signed conjugation, its square, the fixed subgroup and the carrying elements from The Signed Sandwich on a Topological Group and Reflections as Signed Two-Sided Operators on a Topological Group; and the fixed and inverted subgroups and the dictionary from Involutive Topological Groups. The Haar inner product and the adjoint of a reflection under it are Part III.

Throughout, $G$ is a Hausdorff topological group, $k$ is a field of characteristic different from two, $k[G]$ is the group algebra, $\sigma$ is a continuous involution, $\alpha = \sigma\iota$ is the associated continuous involutive automorphism, $Z(G)$ is the centre, and the signed conjugation and the reflection are the operators on $k[G]$

$$ \rho_a = \Sigma^{\alpha}_{a,\,a^{-1}} , \qquad \rho_a(u) = a\,\alpha(u)\,a^{-1} = (c_a\alpha)(u) , $$

the second expression holding by the identification of the linear extension with the automorphism of the group algebra.

The Adjoint of a Reflection

Theorem (the class of reflections is closed under the adjoint). With respect to the signed form $B_\alpha(u,v) = \sum_g u_gv_{\alpha(g)}$, the adjoint of the signed conjugation is the signed conjugation by the inverse,

$$ \rho_a^\dagger = \rho_{a^{-1}} = \Sigma^{\alpha}_{a^{-1},\,a} , $$

so the adjoint of a reflection is the reflection carried by the inverse of the carrying element, and the adjoint operation is an involution on the set of reflections.

Proof. By the adjoint formula for the signed sandwich, $\rho_a^\dagger = (\Sigma^\alpha_{a,a^{-1}})^\dagger = \Sigma^\alpha_{a^{-1},(a^{-1})^{-1}} = \Sigma^\alpha_{a^{-1},a} = \rho_{a^{-1}}$. If $a\alpha(a)$ is central then so is $a^{-1}\alpha(a^{-1}) = (a\alpha(a))^{-1}$, so $\rho_{a^{-1}}$ is again a reflection; and $(\rho_{a^{-1}})^\dagger = \rho_a$.

Corollary (the adjoint in the operator algebra). As an element of the algebra generated by the translations and the linear extension $A$ of the grade involution, $\rho_a = c_aA = L_aAR_{a^{-1}}$, and its adjoint is $A R_{\sigma(a)^{-1}}L_{\sigma(a)^{-1}}$; the two expressions agree because $\sigma(a)^{-1} = \alpha(a)$.

Proof. $\rho_a = L_aR_{a^{-1}}A$, and the adjoint is $A^\dagger R_{a^{-1}}^\dagger L_a^\dagger = A\,R_{\sigma(a^{-1})}\,L_{\sigma(a)}$. Since $\sigma(a^{-1}) = \alpha(a)$ and $\sigma(a)^{-1} = \alpha(a)$ as well, the displayed form follows.

Self-Adjointness and Orthogonality

Theorem (self-adjointness). The signed conjugation $\rho_a$ is self-adjoint, $\rho_a^\dagger = \rho_a$, if and only if $a^2 \in Z(G)$. It is unitary if and only if $a^{-1}\alpha(a) \in Z(G)$. If $\rho_a$ is a reflection — that is, if $a\alpha(a) \in Z(G)$ — then the two conditions are equivalent, so a reflection is self-adjoint exactly when it is unitary, and it is then an orthogonal involution of the group algebra.

Proof. Self-adjointness: $\rho_a^\dagger = \rho_{a^{-1}}$ equals $\rho_a$ exactly when $a^{-1} \in aZ(G)$, by the coset criterion for signed conjugations, and $a^{-1}\in aZ(G)$ is $a^2 \in Z(G)$. Unitarity: by the unitarity criterion with $b = a^{-1}$, the second condition is automatic and the first is $a^{-1}\alpha(a) \in Z(G)$. If $a\alpha(a)$ is central then $\alpha(a) = a^{-1}z$ with $z$ central, so $a^{-1}\alpha(a) = a^{-2}z$, which is central exactly when $a^2$ is central. Finally, an involution is self-adjoint if and only if it is unitary: self-adjointness gives $\rho_a^\dagger\rho_a = \rho_a^2 = \mathrm{id}$, and unitarity with $\rho_a^2 = \mathrm{id}$ gives $\rho_a^\dagger = \rho_a^{-1} = \rho_a$.

Corollary (the involution and its fixed space). Let $\rho_a$ be a self-adjoint reflection. Then the group algebra decomposes as the direct sum of the two eigenspaces of the involution,

$$ k[G] = \ker(\rho_a - \mathrm{id}) \oplus \ker(\rho_a + \mathrm{id}), $$

the two summands are mutually orthogonal for the signed form, and the first is the fixed space of the reflection, the annihilator of the second.

Proof. An involution with $2$ invertible has the decomposition into its $\pm1$-eigenspaces. If $\rho_a$ is self-adjoint and $\rho_au = u$, $\rho_av = -v$ then $B_\alpha(u,v) = B_\alpha(\rho_au,v) = B_\alpha(u,\rho_av) = -B_\alpha(u,v)$, so $B_\alpha(u,v) = 0$; hence the summands are orthogonal and each is the annihilator of the other under the nondegenerate form.

Corollary (the fixed space as a kernel). The fixed space of the reflection is the kernel of the operator $\alpha - c_{a^{-1}}$ on the group algebra, and it is spanned by the elements of the fixed subgroup $\operatorname{Fix}(a) = \{x : \alpha(x) = a^{-1}xa\}$ when the latter is a subgroup; in the topological case the fixed set is closed and the kernel is the closed span.

Proof. $\rho_a u = u$ means $a\alpha(u)a^{-1} = u$, that is $\alpha(u) = a^{-1}ua = c_{a^{-1}}(u)$ on group elements, so $(\alpha - c_{a^{-1}})u = 0$. The solutions in the basis are the elements of $\operatorname{Fix}(a)$, and the kernel is their span, closed by the continuity of the operator.

The Degenerate Case

Theorem (the separation of the conditions). Suppose $a\alpha(a)$ is not central, so that $\rho_a$ is not an involution and not a reflection. Then self-adjointness and unitarity separate: $\rho_a$ is self-adjoint when $a^2$ is central and unitary when $a^{-1}\alpha(a)$ is central, and neither condition implies the other in general. The operator has order the order of the inner conjugation $c_{a\alpha(a)}$, and it is self-adjoint if and only if $\rho_a^{-1} = \rho_a$, a condition on the carrying element alone.

Proof. In the reflection case the centrality of $a\alpha(a)$ made the two conditions equivalent; here it is absent, so the computations $a^2\in Z(G)$ and $a^{-1}\alpha(a)\in Z(G)$ are independent. The order statement is the computation of the square, $\rho_a^2 = c_{a\alpha(a)}$; self-adjointness is $\rho_{a^{-1}} = \rho_a$, which is $a^{-2}\in Z(G)$.

Corollary (the operator with a non-central square). If $a\alpha(a)\in Z(G)$ but $a^2\notin Z(G)$, then $\rho_a$ is an involution of the group algebra that is not self-adjoint: it is an involution whose adjoint is a different involution, $\rho_{a^{-1}}$, and it is not orthogonal. The pair $(\rho_a,\rho_{a^{-1}})$ are mutually adjoint and their product is the identity.

Proof. The square is the identity by the centrality of $a\alpha(a)$, so $\rho_{a^{-1}} = \rho_a^{-1}$; the adjoint is $\rho_{a^{-1}}$, and it differs from $\rho_a$ because $a^2$ is not central, so the involution is not self-adjoint. The product statement is the involution property.

Corollary (the reversion criterion). The reflection $\rho_a$ is orthogonal exactly when $a\,a$ is central, and the orthogonal reflections among the elements of the inverted subgroup $G^\sigma = I(\alpha)$ are those with $a^2$ central; when the action of the adjoint is used to read a reflection as a Hermitian operator, the carrier must satisfy this condition, not merely the involution condition.

Proof. Orthogonality is self-adjointness for an involution, just proved, and the condition is $a^2\in Z(G)$; for $a\in G^\sigma$ one has $\alpha(a) = a^{-1}$, so the involution condition $a\alpha(a)\in Z(G)$ holds automatically, and the orthogonality condition remains $a^2\in Z(G)$.

Summary

Read on the group algebra with respect to the signed form, the signed conjugation $\rho_a$ has adjoint $\rho_{a^{-1}}$, so the class of reflections is closed under the adjoint and the adjoint operation is an involution on it. The signed conjugation is self-adjoint exactly when $a^2$ is central and unitary exactly when $a^{-1}\alpha(a)$ is central; for a genuine reflection the two conditions coincide, and then self-adjointness is the same as unitarity and the reflection is an orthogonal involution. A self-adjoint reflection splits the group algebra into two orthogonal eigenspaces, its fixed space being the kernel of $\alpha - c_{a^{-1}}$ and spanned by the fixed subgroup when the latter is one. In the degenerate case, when $a\alpha(a)$ is not central, the operator is not an involution, the two conditions separate, and a reflection of the group algebra may fail either self-adjointness or unitarity; the involution with a non-central square is the model case of an involution whose adjoint is a different involution. The carrier of an orthogonal reflection is an element whose square is central.

Summary of Notation

Further Reading

• Max-Albert Knus, Alexander Merkurjev, Markus Rost and Jean-Pierre Tignol, The Book of Involutions, Colloquium Publications 44 (American Mathematical Society, 1998), for involutions, adjoints, self-adjoint operators and the orthogonal and unitary groups.
• Ian R. Porteous, Clifford Algebras and the Classical Groups (Cambridge University Press, 1995), for reflections as sandwich operators, their fixed hyperplanes and their orthogonality.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge University Press, second edition, 2001), for the sandwich operator, the reflection and the adjoint in a Clifford algebra.
• Nicolas Bourbaki, Algebra I, Chapters 1–3 (Springer, 1998), for self-adjoint and orthogonal operators with respect to a bilinear form.
• Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis I (Springer, second edition, 1979), for the Haar inner product and the adjoint of a reflection under it, which is Part III and is not used here.