Duvelisib Maintenance After Autologous Stem Cell Transplant in T-Cell Lymphomas

Inclusion Criteria: * Diagnosis of T cell non-Hodgkin lymphoma, T cell lymphomas included are peripheral T cell lymphoma not otherwise specified, angioimmunoblastic T cell lymphoma, and systemic anaplastic large cell lymphoma. * Eligible for autologous stem cell transplantation as determined by the treating physician or completed autologous transplant within the last 30 days. * At least 18 years of age at time of enrollment * Eastern Cooperative Oncology Group (ECOG) performance status ≤ 2 * Adequate organ function as defined below: * Serum creatinine ≤ 1.5 times institutional upper limit of normal (IULN) * Total bilirubin ≤ 1.5 x IULN. Patients with Gilbert's Syndrome may have a bilirubin \> 1.5 x IULN * Hemoglobin ≥ 8.0 g/dL * Absolute neutrophil count ≥ 1.0 x 109/L * Platelet count ≥ 75 x 109/L * AST(SGOT)/ALT(SGPT) ≤ 3.0 x IULN * Women of childbearing potential and men must agree to use highly effective contraception prior to study entry and for the duration of study participation and for 3 months after the last dose of duvelisib. Negative serum β human chorionic gonadotropin (βHCG) pregnancy test within 7 days before first treatment is required if the patient is a woman of childbearing potential. * Participants or a participant's legally authorized representative must be able to understand and willing to sign an IRB approved written informed consent document Exclusion Criteria: * Currently receiving any other experimental therapy or has received experimental therapy within 4 weeks prior to study treatment * History of allergic reaction attributed to compounds of similar chemical or biologic composition to duvelisib or other agents used in the study. * Prior history of drug-induced colitis or drug-induced pneumonitis * History of concurrent interstitial lung disease or severely impaired lung function * History of chronic liver disease or veno-occlusive disease * History of tuberculosis within 2 years prior to enrollment * Administration of a live or live attenuated vaccine within 6 weeks of first duvelisib dose * Ongoing treatment with chronic immunosuppressants (e.g., cyclosporine) or systemic steroids \> 20 mg of prednisone (or equivalent) per day * Ongoing treatment for systemic bacterial, fungal, or viral infections at screening. Note: patients on antimicrobial, antifungal, or antiviral prophylaxis are not specifically excluded is all other inclusion/exclusion criteria are met * Unable to receive prophylactic treatment for pneumocystis, herpes simplex virus (HSV), or herpes zoster (VZV) at screening * Infection with HBV, HCV. Subjects with a positive HBsAg or HCV Ab on pre-transplant infection screening will be excluded. Subjects with a positive HBcAb must have negative HBV DNA to be eligible and must be periodically monitored for HBV reactivation by institutional guidelines. * Baseline QTcF \> 500 milliseconds. This does not apply to subjects with right or left bundle branch blocks * Concurrent active malignancy other than non-melanoma skin cancer or carcinoma in situ of the cervix, bladder cancer, or prostate cancer not requiring treatment. * Clinically significant medical condition of malabsorption, inflammatory bowel disease, chronic conditions which manifest with diarrhea, refractory nausea,vomiting, or any other condition that will interfere significantly with drug absorption * Concurrent administration of medications or foods that are strong inhibitors or inducers of cytochrome P450 3A (CYP3A). No prior use within 2 weeks before the start of study intervention. * Active cytomegalovirus (CMV) or Epstein-Barr virus (EBV) infection (i.e., subjects with detectable viral load) * History of stroke, unstable angina, myocardial infarction, or ventricular arrhythmia requiring medication or a pacemaker within the last 6 months prior to screening. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, or unstable cardiac arrhythmia. * Pregnant or breastfeeding. * Patients with HIV are eligible unless their CD4+ T-cell counts are \< 350 cells/mcL or they have a history of AIDS-defining opportunistic infection within the 12 months prior to registration. Concurrent treatment with effective ART according to DHHS treatment guidelines is recommended. Recommend exclusion of specific ART agents based on predicted drug-drug interactions (i.e. concurrent strong CYP3A4 inhibitors (ritonavir and cobicistat) or inducers (efavirenz) are contraindicated).