Reductive Subgroup Research Articles

Response to duloxetine in patients with knee pain due to osteoarthritis: an exploratory post hoc analysis of a Japanese Phase III randomized study.

PurposeTo assess whether patients with knee osteoarthritis pain who have early pain reduction or treatment-related adverse events of special interest (TR-AESIs; constipation, decreased appetite, malaise, nausea, somnolence, thirst) with duloxetine treatment are more likely to have later improvements in pain and quality of life (QOL) relative to placebo than patients without these early indicators.Patients and methodsThis was a post hoc analysis of 14-week randomized trial of Japanese patients with knee osteoarthritis pain (Brief Pain Inventory [BPI]-Severity average pain score ≥4) receiving duloxetine 60 mg/day (n=177 analyzed) or placebo (n=176). Primary trial outcome was change from baseline in BPI-Severity average pain at Week 14. Subgroups included early pain reduction (≥30%, 10%–30%, or <10% decrease in BPI-Severity average pain at Week 4) and early TR-AESIs (with/without TR-AESIs by Week 2). Measures included changes from baseline in BPI-Severity average pain, QOL (BPI-Interference, Western Ontario and McMaster Universities Osteoarthritis Index), Patient Global Impression of Improvement (PGI-I), and response rate (proportion achieving ≥30% or ≥50% pain reduction at Week 14).ResultsThe ≥30% early pain reduction subgroup (n=93) had significantly greater improvements in pain, QOL, and PGI-I and higher ≥30% and ≥50% response rates than placebo; the 10%–30% (n=45) and the <10% (n=33) pain reduction subgroups did not show the same (except 10%–30% group: PGI-I at Week 10 and some QOL at Weeks 10 and/or 14). Both TR-AESI subgroups (with, n=52; without, n=125) had significantly greater improvements in pain, PGI-I, and most QOL measures and higher response rates than placebo.ConclusionEarly efficacy responses to duloxetine treatment, but not early TR-AESIs, may predict later pain reduction and QOL improvements in Japanese patients with knee osteoarthritis pain.ClinicalTrials.govNCT02248480.

Reducible subgroups of exceptional algebraic groups

Let G be a simple algebraic group over an algebraically closed field. A closed subgroup H of G is called G-completely reducible (G-cr) if, whenever H is contained in a parabolic subgroup P of G, it is contained in a Levi factor of P. In this paper we complete the classification of connected G-cr subgroups when G has exceptional type, by determining the L0-irreducible connected reductive subgroups for each simple classical factor L0 of a Levi subgroup of G. As an illustration, we determine all reducible, G-cr semisimple subgroups when G has type F4 and various properties thereof. This work complements results of Lawther, Liebeck, Seitz and Testerman, and is vital in classifying non-G-cr reductive subgroups, a project being undertaken by the authors elsewhere.

Semisimple weakly symmetric pseudo-Riemannian manifolds

We develop the classification of weakly symmetric pseudo-Riemannian manifolds G / H where G is a semisimple Lie group and H is a reductive subgroup. We derive the classification from the cases where G is compact, and then we discuss the (isotropy) representation of H on the tangent space of G / H and the signature of the invariant pseudo-Riemannian metric. As a consequence we obtain the classification of semisimple weakly symmetric manifolds of Lorentz signature $$(n-1,1)$$ and trans-Lorentzian signature $$(n-2,2)$$ .

Complete reducibility in good characteristic

Let G G be a simple algebraic group of exceptional type, over an algebraically closed field of characteristic p ≥ 0 p \ge 0 . A closed subgroup H H of G G is called G G -completely reducible ( G G -cr) if whenever H H is contained in a parabolic subgroup P P of G G , it is contained in a Levi subgroup of P P . In this paper we determine the G G -conjugacy classes of non- G G -cr simple connected subgroups of G G when p p is good for G G . For each such subgroup X X , we determine the action of X X on the adjoint module L ( G ) L(G) and the connected centraliser of X X in G G . As a consequence we classify all non- G G -cr connected reductive subgroups of G G , and determine their connected centralisers. We also classify the subgroups of G G which are maximal among connected reductive subgroups, but not maximal among all connected subgroups.

Unipotent elements forcing irreducibility in linear algebraic groups

Abstract Let G be a simple algebraic group over an algebraically closed field K of characteristic p &gt; 0 {p&gt;0} . We consider connected reductive subgroups X of G that contain a given distinguished unipotent element u of G. A result of Testerman and Zalesski [D. Testerman and A. Zalesski, Irreducibility in algebraic groups and regular unipotent elements, Proc. Amer. Math. Soc. 141 2013, 1, 13–28] shows that if u is a regular unipotent element, then X cannot be contained in a proper parabolic subgroup of G. We generalize their result and show that if u has order p, then except for two known examples which occur in the case ( G , p ) = ( C 2 , 2 ) {(G,p)=(C_{2},2)} , the subgroup X cannot be contained in a proper parabolic subgroup of G. In the case where u has order &gt; p {&gt;p} , we also present further examples arising from indecomposable tilting modules with quasi-minuscule highest weight.

Classification of Klein four symmetric pairs of holomorphic type for $$\mathrm {E}_{6(-14)}$$ E 6 ( - 14 )

The author classifies Klein four symmetric pairs of holomorphic type for non-compact Lie group $\mathrm{E}_{6(-14)}$, which gives a class of pairs $(G,G')$ of real reductive Lie group $G$ and its reductive subgroup $G'$ such that there exist irreducible unitary representations $\pi$ of $G$, which are admissible upon restriction to $G'$.

AR-Components of domestic finite group schemes: McKay-Quivers and Ramification

For a domestic finite group scheme, we give a direct description of the Euclidean components in its Auslander-Reiten quiver via the McKay-quiver of a finite linearly reductive subgroup scheme of $SL(2)$. Moreover, for a normal subgroup scheme $\mathcal{N}$ of a finite group scheme $\mathcal{G}$, we show that there is a connection between the ramification indices of the restriction morphism $\mathbb{P}(\mathcal{V}_{\mathcal{N}})\rightarrow\mathbb{P}(\mathcal{V}_{\mathcal{G}})$ between their projectivized cohomological support varieties and the ranks of the tubes in their Auslander-Reiten quivers.

Coadjoint geometry for discretely decomposable restrictions of certain series of representations of indefinite unitary groups

Zuckerman’s derived functor module of a semisimple Lie group [Formula: see text] yields a unitary representation [Formula: see text] which may be regarded as a geometric quantization of an elliptic orbit [Formula: see text] in the Kirillov–Kostant–Duflo orbit philosophy. We highlight a certain family of those irreducible unitary representations [Formula: see text] of the indefinite unitary group [Formula: see text] and a family of subgroups [Formula: see text] of [Formula: see text] such that the restriction [Formula: see text] is known to be discretely decomposable and multiplicity-free by the general theory of Kobayashi (Discrete decomposibility of the restrictions of [Formula: see text] with respect to reductive subgroups, II, Ann. of Math. 147 (1998) 1–21; Multiplicity-free representations and visible action on complex manifolds, Publ. Res. Inst. Math. Sci. 41 (2005) 497–549), where [Formula: see text] is not necessarily tempered and [Formula: see text] is not necessarily compact. We prove that the corresponding moment map [Formula: see text] is proper, determine the image [Formula: see text], and compute the Corwin–Greenleaf multiplicity function explicitly.

On simple polynomial GrT-modules

Using the general framework of polynomial representations defined by Doty and generalizing the definition given by Doty, Nakano and Peters for G=GLn, we consider polynomial representations of GrT for an arbitrary closed reductive subgroup scheme G⊆GLn and a maximal torus T of G in positive characteristic. We give sufficient conditions on G making a classification of simple polynomial GrT-modules similar to the case G=GLn possible and apply this to recover the corresponding result for GLn with a different proof, extending it to symplectic similitude groups, Levi subgroups of GLn and, in a weaker form, to odd orthogonal similitude groups. We also consider orbits of the affine Weyl group and give a condition for equivalence of blocks of polynomial representations for GrT in the case G=GLn.

SUFFICIENT CONDITIONS FOR HOLOMORPHIC LINEARISATION

Let G be a reductive complex Lie group acting holomorphically on X = ℂn. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ℂn such that the G-action becomes linear. Equivalently, is there a G-equivariant biholomorphism Φ: X → V where V is a G-module? There is an intrinsic stratification of the categorical quotient QX, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of G. Suppose that there is a Φ as above. Then Φ induces a biholomorphism φ: QX → QV which is stratified, i.e., the stratum of QX with a given label is sent isomorphically to the stratum of QV with the same label.

Which factor is most important for occurrence of cutout complications in patients treated with proximal femoral nail antirotation? Retrospective analysis of 298 patients.

Mechanical complications, such as cut-out of the head-neck fixation device, are the most common causes of morbidity after trochanteric femur fracture treatment. The causes of cut-out complications are well defined in patients who are treated with sliding hip screws and biaxial cephalomedullary nails but there are few reports about the patients who are treated with proximal femoral nail antirotation. The purpose of this study was to evaluate the most important factor about occurance of cutout complication and also to evaluate the risks of the combination of each possible factors. Overally 298 patients were enrolled in the study. Medical records were reviewed for patients' age, fracture type, gender, anesthesia type and occurance of cut-out complication. Postoperatively taken radiographs were reviewed for tip-apex distance, obtained collo-diaphyseal angle, the quadrant of the helical blade and Ikuta reduction subgroup. The most important factor (s) and also predicted probability of cut-out complication was calculated for each combination of factors. Cut-out complication was observed in 14 patients (4.7%). The most important factor about occurrence of the cut-out complication was found as varus reduction (p: 0.01), the second important factor was found as implantation of the helical blade in the improper quadrant (p: 0.02). Tip-apex distance was found as third important factor (p: 0.10). The predicted probability of cut-out complication was calculated as 45.6% when whole of the four surgeon dependent factors were improperly obtained. Althought obtaining proper tip-apex distance is important to prevent cutout complication in these fractures, if the fracture is not reduced in varus position and helical blade is inserted in the proper quadrant, possibility of cut-out complication is very low even in the patients with high tip-apex distance.

ON ALGEBRAIC VOLUME DENSITY PROPERTY

A smooth affine algebraic variety X equipped with an algebraic volume form ω has the algebraic volume density property (AVDP) if the Lie algebra generated by complete algebraic vector fields of ω-divergence zero coincides with the space of all algebraic vector fields of ω-divergence zero. We develop an effective criterion of verifying whether a given X has AVDP. As an application of this method we establish AVDP for any homogeneous space X = G/R that admits a G-invariant algebraic volume form where G is a linear algebraic group and R is a closed reductive subgroup of G.

Properly discontinuous group actions on affine homogeneous spaces

Let G be a real algebraic group, H ≤ G an algebraic subgroup containing a maximal reductive subgroup of G, and Γ a subgroup of G acting on G/H by left translations. We conjecture that Γ is virtually solvable provided its action on G/H is properly discontinuous and ΓG/H is compact, and we confirm this conjecture when G does not contain simple algebraic subgroups of rank ≥2. If the action of Γ on G/H (which is isomorphic to an affine linear space A n ) is linear, our conjecture coincides with the Auslander conjecture. We prove the Auslander conjecture for n ≤ 5.

Tempered reductive homogeneous spaces

Let G be a semisimple algebraic Lie group and H a reductive subgroup. We find geometrically the best even integer p for which the representation of G in L^2(G/H) is almost L^{p}. As an application, we give a criterion which detects whether this representation is tempered.

On the restriction of Zuckerman’s derived functor modules A q (λ) to reductive subgroups

In this article, we study the restriction of Zuckerman's derived functor $({\frak g},K)$-modules $A_{\frak q}}(\lambda)$ to ${\frak g}'$ for symmetric pairs of reductive Lie algebras $({\frak g},{\frak g}')$. When the restriction decomposes into irreducible $({\frak g}',K')$-modules, we give an upper bound for the branching law. In particular, we prove that each $({\frak g}',K')$-module occurring in the restriction is isomorphic to a submodule of $A_{\frak q}'}(\lambda')$ for a parabolic subalgebra ${\frak q}'$ of ${\frak g}'$, and determine their associated varieties. For the proof, we realize $A_{\frak q}}(\lambda)$ on complex partial flag varieties by using ${\mathcal D}$-modules.

Quantization commutes with reduction in the non-compact setting: the case of holomorphic discrete series

In this paper we show that the multiplicities of holomorphic discrete series representations relative to reductive subgroups satisfy the credo “quantization commutes with reduction”.

A restriction on proper group actions on homogeneous spaces of reductive type

Let $$L$$ be a reductive subgroup of a reductive Lie group $$G$$ . Let $$G/H$$ be a homogeneous space of reductive type. We provide a necessary condition for the properness of the action of $$L$$ on $$G/H$$ . As an application we give examples of spaces that do not admit standard compact Clifford-Klein forms.

Crystallographic actions on contractible algebraic manifolds

We study properly discontinuous and cocompact actions of a discrete subgroup Γ \Gamma of an algebraic group G G on a contractible algebraic manifold X X . We suppose that this action comes from an algebraic action of G G on X X such that a maximal reductive subgroup of G G fixes a point. When the real rank of any simple subgroup of G G is at most one or the dimension of X X is at most three, we show that Γ \Gamma is virtually polycyclic. When Γ \Gamma is virtually polycyclic, we show that the action reduces to an NIL-affine crystallographic action. Specializing to NIL-affine actions, we prove that the generalized Auslander conjecture holds up to dimension six and give a new proof of the fact that every virtually polycyclic group admits an NIL-affine crystallographic action.

Efficacy of long-term milnacipran treatment in patients meeting different thresholds of clinically relevant pain relief: subgroup analysis of a randomized, double-blind, placebo-controlled withdrawal study.

BackgroundFibromyalgia patients from a long-term, open-label study of milnacipran (50–200 mg/day) were eligible to participate in a 12-week, randomized, placebo-controlled withdrawal study. The withdrawal study evaluated loss of therapeutic response in patients who achieved ≥50% pain improvements after receiving up to 3.25 years of milnacipran. This post-hoc analysis investigated whether patients who met lower thresholds of pain improvement also experienced worsening of fibromyalgia symptoms upon treatment withdrawal.MethodAmong patients who received milnacipran ≥100 mg/day during the long-term study, three subgroups were identified based on percentage of pain reduction at randomization: ≥50% (protocol-defined “responders”; n=150); ≥30% to <50% (patients with clinically meaningful pain improvement; n=61); and <30% (n=110). Efficacy assessments included the visual analog scale (VAS) for pain, Fibromyalgia Impact Questionnaire-Revised (FIQR), 36-Item Short-Form Health Survey Physical Component Summary (SF-36 PCS), and Beck Depression Inventory (BDI).ResultsIn the ≥30 to <50% subgroup, significant worsening in pain was detected after treatment withdrawal. The difference between placebo and milnacipran in mean VAS score changes for this subgroup (+9.0, P<0.05) was similar to the difference in protocol-defined responders (+9.4, P<0.05). In the <30% subgroup, no worsening in pain was observed in either treatment arm. However, patients in this subgroup experienced significant worsening in FIQR scores after treatment withdrawal (placebo, +6.9; milnacipran, −2.8; P<0.001), as well as worsening in SF-36 PCS and BDI scores.ConclusionPatients who experienced ≥30% to <50% pain reduction with long-term milnacipran had significant worsening of fibromyalgia symptoms after treatment withdrawal. These results suggest that the conventional ≥30% pain responder cutoff may be adequate to demonstrate efficacy in randomized withdrawal studies of fibromyalgia. Patients in the <30% pain reduction subgroup had worsening scores on the FIQR and other multidimensional measures after treatment withdrawal, indicating the importance of identifying and managing the multiple symptoms of fibromyalgia.

STABILITY OF BRANCHING LAWS FOR HIGHEST WEIGHT MODULES

In this paper, we study the irreducible decomposition of a (ℂ[X];G)-module M for a quasi-affine spherical variety X of a connected reductive algebraic group G over ℂ. We show that for sufficiently large parameters, the decomposition of M with respect to G is reduced to the decomposition of the ‘fiber’ M/\( \mathfrak{m} \)(x 0)M with respect to some reductive subgroup L of G. In particular, we obtain a method to compute the maximum value of multiplicities in M. Our main result is a generalization of earlier work by F. Satō in [17]. We apply this result to branching laws of holomorphic discrete series representations with respect to symmetric pairs of holomorphic type. We give a necessary and sufficient condition for multiplicity-freeness of the branching laws.