THE DISTRICT'S public housing system -- often criticized for mismanagement, lethargy and deficient accountability -- will receive some high level federal scrutiny from the Department of Housing and Urban Development. It apparently is badly needed. HUD Secretary Samuel R. Pierce has ordered his acting assistant secretary for public and Indian housing to work personally with the District government to shape up the city's subsidized housing.

HUD is not taking over the city's public housing system, and Mr. Pierce and Mayor Marion Barry take pains to say that this action is mild. But the District government should find little solace in that. It is still an extraordinary sign that HUD does not believe the city's Housing and Community Development Department has resolved many of the problems disclosed in a 1984 HUD audit. HUD is also concerned with more recent reports of other problems.

Federal officials say it should take no more than 10 days to put a vacated public housing unit, needing minor to moderate repairs, back on the ready- to-be-rented list. They have found that, on average, it takes as long as 20 months to get such units ready in Washington, a city where the waiting list for public housing is five years long.

HUD officials say they have warned the District's housing department "for several years" that it was violating federal regulations by not reimbursing public assistance tenants for utility costs with the federal money for which they were eligible to defray winter heating bills.

The sluggishness with which the District used federal money from the Comprehensive Improvement Assistance Program, which gives cities federal funds to renovate and modernize public housing units, was cited by HUD officials as a reason that the District received no such funds in 1985.

More recently, public housing tenants have complained that it takes weeks or months for the city to respond to reports of squalid conditions. There have been reports that the city cannot fully account for about $1 million in major appliances that were purchased for its public housing units. There have also been accounts of alcohol and drug abuse by maintenance staff members.

If the decision to send a high-ranking HUD official to work with the District has the effect of making city officials more diligent, then that will benefit the residents who inhabit -- or wait to be able to move into -- public housing. The District ranks 10th in the nation in the number of public housing units within its boundaries. Its neighbor, Baltimore, ranks fourth and is regularly cited by HUD officials for the good management of its public housing. District residents deserve the same level of good management. They do not have it now.